← Question derivation index · Original question batch
R0054-27725 · q-20261015-advance-20261118-r04-wilson-theorem-0100
Use mode=fermat_confusion and exact integer n=97. Give (mode, n, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, Wilson_statement, Fermat_statement, theorem_identification_claim, boundary).
G001 — Use mode=fermat_confusion and exact integer n=97. Give (mode, n, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, Wilson_statement, Fermat_statement, theorem_identification_claim, boundary).Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
source roles = [wilson-boundary]
After
derived = [[mode, n, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, Wilson_statement, Fermat_statement, theorem_identification_claim, boundary]]
Conditions: none
List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product1 := mul(1, 1)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product1 = 1
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 1, 1, 1]
After
derived = [0, 1, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S002
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product2 := mul(1, 2)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product2 = 2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 2, 1, 2]
After
derived = [0, 2, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S004
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product3 := mul(2, 3)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product3 = 6
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 3, 2, 6]
After
derived = [0, 6, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S006
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product4 := mul(6, 4)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product4 = 24
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 4, 6, 24]
After
derived = [0, 24, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S008
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product5 := mul(24, 5)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product5 = 120
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 5, 24, 120]
After
derived = [1, 23, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S010
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 := mul(23, 6)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 = 138
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 6, 23, 138]
After
derived = [1, 41, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S012
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 := mul(41, 7)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 = 287
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 7, 41, 287]
After
derived = [2, 93, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S014
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 := mul(93, 8)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 = 744
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 8, 93, 744]
After
derived = [7, 65, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S016
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 := mul(65, 9)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 = 585
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 9, 65, 585]
After
derived = [6, 3, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S018
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 := mul(3, 10)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 = 30
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 10, 3, 30]
After
derived = [0, 30, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S020
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 := mul(30, 11)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 = 330
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 11, 30, 330]
After
derived = [3, 39, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S022
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 := mul(39, 12)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 = 468
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 12, 39, 468]
After
derived = [4, 80, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S024
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 := mul(80, 13)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 = 1040
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 13, 80, 1040]
After
derived = [10, 70, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S026
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 := mul(70, 14)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 = 980
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 14, 70, 980]
After
derived = [10, 10, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S028
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 := mul(10, 15)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 = 150
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 15, 10, 150]
After
derived = [1, 53, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S030
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 := mul(53, 16)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 = 848
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 16, 53, 848]
After
derived = [8, 72, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S032
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 := mul(72, 17)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 = 1224
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 17, 72, 1224]
After
derived = [12, 60, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S034
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 := mul(60, 18)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 = 1080
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 18, 60, 1080]
After
derived = [11, 13, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S036
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 := mul(13, 19)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 = 247
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 19, 13, 247]
After
derived = [2, 53, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S038
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 := mul(53, 20)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 = 1060
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 20, 53, 1060]
After
derived = [10, 90, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S040
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 := mul(90, 21)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 = 1890
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 21, 90, 1890]
After
derived = [19, 47, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S042
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 := mul(47, 22)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 = 1034
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 22, 47, 1034]
After
derived = [10, 64, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S044
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 := mul(64, 23)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 = 1472
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 23, 64, 1472]
After
derived = [15, 17, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S046
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 := mul(17, 24)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 = 408
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 24, 17, 408]
After
derived = [4, 20, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S048
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 := mul(20, 25)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 = 500
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 25, 20, 500]
After
derived = [5, 15, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S050
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 := mul(15, 26)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 = 390
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 26, 15, 390]
After
derived = [4, 2, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S052
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 := mul(2, 27)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 = 54
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 27, 2, 54]
After
derived = [0, 54, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S054
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 := mul(54, 28)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 = 1512
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 28, 54, 1512]
After
derived = [15, 57, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S056
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 := mul(57, 29)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 = 1653
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 29, 57, 1653]
After
derived = [17, 4, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S058
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 := mul(4, 30)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 = 120
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 30, 4, 120]
After
derived = [1, 23, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S060
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 := mul(23, 31)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 = 713
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 31, 23, 713]
After
derived = [7, 34, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S062
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 := mul(34, 32)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 = 1088
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S063
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 32, 34, 1088]
After
derived = [11, 21, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S064
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 := mul(21, 33)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 = 693
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S065
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 33, 21, 693]
After
derived = [7, 14, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S066
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 := mul(14, 34)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 = 476
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S067
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 34, 14, 476]
After
derived = [4, 88, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S068
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 := mul(88, 35)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 = 3080
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S069
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 35, 88, 3080]
After
derived = [31, 73, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S070
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 := mul(73, 36)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 = 2628
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S071
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 36, 73, 2628]
After
derived = [27, 9, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S072
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 := mul(9, 37)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 = 333
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S073
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 37, 9, 333]
After
derived = [3, 42, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S074
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 := mul(42, 38)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 = 1596
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S075
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 38, 42, 1596]
After
derived = [16, 44, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S076
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 := mul(44, 39)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 = 1716
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S077
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 39, 44, 1716]
After
derived = [17, 67, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S078
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 := mul(67, 40)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 = 2680
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S079
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 40, 67, 2680]
After
derived = [27, 61, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S080
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 := mul(61, 41)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 = 2501
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S081
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 41, 61, 2501]
After
derived = [25, 76, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S082
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 := mul(76, 42)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 = 3192
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S083
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 42, 76, 3192]
After
derived = [32, 88, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S084
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product43 := mul(88, 43)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product43 = 3784
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S085
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 43, 88, 3784]
After
derived = [39, 1, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S086
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 := mul(1, 44)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 = 44
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S087
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 44, 1, 44]
After
derived = [0, 44, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S088
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 := mul(44, 45)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 = 1980
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S089
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 45, 44, 1980]
After
derived = [20, 40, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S090
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product46 := mul(40, 46)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product46 = 1840
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S091
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 46, 40, 1840]
After
derived = [18, 94, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S092
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 := mul(94, 47)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 = 4418
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S093
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 47, 94, 4418]
After
derived = [45, 53, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S094
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 := mul(53, 48)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 = 2544
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S095
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 48, 53, 2544]
After
derived = [26, 22, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S096
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product49 := mul(22, 49)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product49 = 1078
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S097
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 49, 22, 1078]
After
derived = [11, 11, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S098
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product50 := mul(11, 50)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product50 = 550
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S099
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 50, 11, 550]
After
derived = [5, 65, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S100
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product51 := mul(65, 51)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product51 = 3315
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S101
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 51, 65, 3315]
After
derived = [34, 17, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S102
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product52 := mul(17, 52)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product52 = 884
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S103
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 52, 17, 884]
After
derived = [9, 11, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S104
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product53 := mul(11, 53)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product53 = 583
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S105
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 53, 11, 583]
After
derived = [6, 1, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S106
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product54 := mul(1, 54)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product54 = 54
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S107
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 54, 1, 54]
After
derived = [0, 54, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S108
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product55 := mul(54, 55)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product55 = 2970
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S109
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 55, 54, 2970]
After
derived = [30, 60, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S110
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product56 := mul(60, 56)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product56 = 3360
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S111
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 56, 60, 3360]
After
derived = [34, 62, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S112
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product57 := mul(62, 57)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product57 = 3534
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S113
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 57, 62, 3534]
After
derived = [36, 42, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S114
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product58 := mul(42, 58)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product58 = 2436
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S115
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 58, 42, 2436]
After
derived = [25, 11, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S116
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product59 := mul(11, 59)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product59 = 649
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S117
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 59, 11, 649]
After
derived = [6, 67, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S118
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product60 := mul(67, 60)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product60 = 4020
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S119
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 60, 67, 4020]
After
derived = [41, 43, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S120
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product61 := mul(43, 61)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product61 = 2623
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S121
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 61, 43, 2623]
After
derived = [27, 4, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S122
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product62 := mul(4, 62)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product62 = 248
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S123
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 62, 4, 248]
After
derived = [2, 54, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S124
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product63 := mul(54, 63)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product63 = 3402
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S125
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 63, 54, 3402]
After
derived = [35, 7, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S126
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product64 := mul(7, 64)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product64 = 448
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S127
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 64, 7, 448]
After
derived = [4, 60, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S128
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product65 := mul(60, 65)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product65 = 3900
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S129
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 65, 60, 3900]
After
derived = [40, 20, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S130
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product66 := mul(20, 66)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product66 = 1320
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S131
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 66, 20, 1320]
After
derived = [13, 59, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S132
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product67 := mul(59, 67)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product67 = 3953
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S133
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 67, 59, 3953]
After
derived = [40, 73, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S134
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product68 := mul(73, 68)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product68 = 4964
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S135
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 68, 73, 4964]
After
derived = [51, 17, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S136
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product69 := mul(17, 69)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product69 = 1173
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S137
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 69, 17, 1173]
After
derived = [12, 9, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S138
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product70 := mul(9, 70)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product70 = 630
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S139
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 70, 9, 630]
After
derived = [6, 48, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S140
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product71 := mul(48, 71)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product71 = 3408
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S141
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 71, 48, 3408]
After
derived = [35, 13, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S142
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product72 := mul(13, 72)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product72 = 936
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S143
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 72, 13, 936]
After
derived = [9, 63, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S144
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product73 := mul(63, 73)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product73 = 4599
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S145
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 73, 63, 4599]
After
derived = [47, 40, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S146
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product74 := mul(40, 74)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product74 = 2960
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S147
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 74, 40, 2960]
After
derived = [30, 50, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S148
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product75 := mul(50, 75)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product75 = 3750
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S149
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 75, 50, 3750]
After
derived = [38, 64, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S150
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product76 := mul(64, 76)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product76 = 4864
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S151
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 76, 64, 4864]
After
derived = [50, 14, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S152
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product77 := mul(14, 77)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product77 = 1078
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S153
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 77, 14, 1078]
After
derived = [11, 11, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S154
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product78 := mul(11, 78)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product78 = 858
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S155
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 78, 11, 858]
After
derived = [8, 82, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S156
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product79 := mul(82, 79)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product79 = 6478
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S157
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 79, 82, 6478]
After
derived = [66, 76, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S158
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product80 := mul(76, 80)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product80 = 6080
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S159
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 80, 76, 6080]
After
derived = [62, 66, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S160
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product81 := mul(66, 81)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product81 = 5346
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S161
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 81, 66, 5346]
After
derived = [55, 11, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S162
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product82 := mul(11, 82)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product82 = 902
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S163
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 82, 11, 902]
After
derived = [9, 29, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S164
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product83 := mul(29, 83)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product83 = 2407
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S165
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 83, 29, 2407]
After
derived = [24, 79, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S166
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product84 := mul(79, 84)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product84 = 6636
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S167
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 84, 79, 6636]
After
derived = [68, 40, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S168
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product85 := mul(40, 85)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product85 = 3400
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S169
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 85, 40, 3400]
After
derived = [35, 5, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S170
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product86 := mul(5, 86)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product86 = 430
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S171
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 86, 5, 430]
After
derived = [4, 42, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S172
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product87 := mul(42, 87)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product87 = 3654
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S173
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 87, 42, 3654]
After
derived = [37, 65, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S174
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product88 := mul(65, 88)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product88 = 5720
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S175
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 88, 65, 5720]
After
derived = [58, 94, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S176
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product89 := mul(94, 89)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product89 = 8366
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S177
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 89, 94, 8366]
After
derived = [86, 24, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S178
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product90 := mul(24, 90)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product90 = 2160
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S179
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 90, 24, 2160]
After
derived = [22, 26, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S180
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product91 := mul(26, 91)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product91 = 2366
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S181
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 91, 26, 2366]
After
derived = [24, 38, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S182
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product92 := mul(38, 92)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product92 = 3496
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S183
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 92, 38, 3496]
After
derived = [36, 4, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S184
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product93 := mul(4, 93)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product93 = 372
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S185
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 93, 4, 372]
After
derived = [3, 81, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S186
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product94 := mul(81, 94)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product94 = 7614
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S187
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 94, 81, 7614]
After
derived = [78, 48, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S188
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product95 := mul(48, 95)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product95 = 4560
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S189
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 95, 48, 4560]
After
derived = [47, 1, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S190
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product96 := mul(1, 96)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product96 = 96
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S191
Before
source roles = [Wilson boundary ORIGINAL prime, 97, 96, 1, 96]
After
derived = [0, 96, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S192
Before
source roles = [97, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 23], [6, 41], [7, 93], [8, 65], [9, 3], [10, 30], [11, 39], [12, 80], [13, 70], [14, 10], [15, 53], [16, 72], [17, 60], [18, 13], [19, 53], [20, 90], [21, 47], [22, 64], [23, 17], [24, 20], [25, 15], [26, 2], [27, 54], [28, 57], [29, 4], [30, 23], [31, 34], [32, 21], [33, 14], [34, 88], [35, 73], [36, 9], [37, 42], [38, 44], [39, 67], [40, 61], [41, 76], [42, 88], [43, 1], [44, 44], [45, 40], [46, 94], [47, 53], [48, 22], [49, 11], [50, 65], [51, 17], [52, 11], [53, 1], [54, 54], [55, 60], [56, 62], [57, 42], [58, 11], [59, 67], [60, 43], [61, 4], [62, 54], [63, 7], [64, 60], [65, 20], [66, 59], [67, 73], [68, 17], [69, 9], [70, 48], [71, 13], [72, 63], [73, 40], [74, 50], [75, 64], [76, 14], [77, 11], [78, 82], [79, 76], [80, 66], [81, 11], [82, 29], [83, 79], [84, 40], [85, 5], [86, 42], [87, 65], [88, 94], [89, 24], [90, 26], [91, 38], [92, 4], [93, 81], [94, 48], [95, 1], [96, 96]], 96]
After
derived = [96, Yes, No]
Conditions: none
Wilson concerns(n-1)!;Fermat concerns a^(n-1) when n does not divide a.This branch supplies no a or power computation.The named comparison neither identifies the theorems nor adds an executed power test.
Rule: Wilson_and_Fermat_DISTINCT_named_statements_NO_unprovided_a_power · Uses: G001, S193
S194 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S193 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S192 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product96 with exact rational arithmetic, not rounded decimals.S191 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S190 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product95 with exact rational arithmetic, not rounded decimals.S189 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S188 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product94 with exact rational arithmetic, not rounded decimals.S187 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S186 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product93 with exact rational arithmetic, not rounded decimals.S185 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S184 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product92 with exact rational arithmetic, not rounded decimals.S183 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S182 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product91 with exact rational arithmetic, not rounded decimals.S181 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S180 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product90 with exact rational arithmetic, not rounded decimals.S179 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S178 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product89 with exact rational arithmetic, not rounded decimals.S177 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S176 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product88 with exact rational arithmetic, not rounded decimals.S175 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S174 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product87 with exact rational arithmetic, not rounded decimals.S173 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S172 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product86 with exact rational arithmetic, not rounded decimals.S171 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S170 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product85 with exact rational arithmetic, not rounded decimals.S169 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S168 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product84 with exact rational arithmetic, not rounded decimals.S167 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S166 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product83 with exact rational arithmetic, not rounded decimals.S165 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S164 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product82 with exact rational arithmetic, not rounded decimals.S163 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S162 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product81 with exact rational arithmetic, not rounded decimals.S161 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S160 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product80 with exact rational arithmetic, not rounded decimals.S159 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S158 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product79 with exact rational arithmetic, not rounded decimals.S157 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S156 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product78 with exact rational arithmetic, not rounded decimals.S155 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S154 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product77 with exact rational arithmetic, not rounded decimals.S153 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S152 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product76 with exact rational arithmetic, not rounded decimals.S151 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S150 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product75 with exact rational arithmetic, not rounded decimals.S149 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S148 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product74 with exact rational arithmetic, not rounded decimals.S147 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S146 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product73 with exact rational arithmetic, not rounded decimals.S145 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S144 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product72 with exact rational arithmetic, not rounded decimals.S143 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S142 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product71 with exact rational arithmetic, not rounded decimals.S141 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S140 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product70 with exact rational arithmetic, not rounded decimals.S139 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S138 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product69 with exact rational arithmetic, not rounded decimals.S137 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S136 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product68 with exact rational arithmetic, not rounded decimals.S135 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S134 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product67 with exact rational arithmetic, not rounded decimals.S133 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S132 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product66 with exact rational arithmetic, not rounded decimals.S131 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S130 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product65 with exact rational arithmetic, not rounded decimals.S129 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S128 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product64 with exact rational arithmetic, not rounded decimals.S127 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S126 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product63 with exact rational arithmetic, not rounded decimals.S125 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S124 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product62 with exact rational arithmetic, not rounded decimals.S123 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S122 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product61 with exact rational arithmetic, not rounded decimals.S121 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S120 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product60 with exact rational arithmetic, not rounded decimals.S119 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S118 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product59 with exact rational arithmetic, not rounded decimals.S117 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S116 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product58 with exact rational arithmetic, not rounded decimals.S115 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S114 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product57 with exact rational arithmetic, not rounded decimals.S113 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S112 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product56 with exact rational arithmetic, not rounded decimals.S111 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S110 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product55 with exact rational arithmetic, not rounded decimals.S109 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S108 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product54 with exact rational arithmetic, not rounded decimals.S107 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S106 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product53 with exact rational arithmetic, not rounded decimals.S105 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S104 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product52 with exact rational arithmetic, not rounded decimals.S103 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S102 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product51 with exact rational arithmetic, not rounded decimals.S101 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S100 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product50 with exact rational arithmetic, not rounded decimals.S099 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S098 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product49 with exact rational arithmetic, not rounded decimals.S097 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S096 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 with exact rational arithmetic, not rounded decimals.S095 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S094 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 with exact rational arithmetic, not rounded decimals.S093 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S092 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product46 with exact rational arithmetic, not rounded decimals.S091 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S090 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 with exact rational arithmetic, not rounded decimals.S089 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S088 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 with exact rational arithmetic, not rounded decimals.S087 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S086 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product43 with exact rational arithmetic, not rounded decimals.S085 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S084 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 with exact rational arithmetic, not rounded decimals.S083 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S082 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 with exact rational arithmetic, not rounded decimals.S081 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S080 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 with exact rational arithmetic, not rounded decimals.S079 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S078 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 with exact rational arithmetic, not rounded decimals.S077 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S076 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 with exact rational arithmetic, not rounded decimals.S075 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S074 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 with exact rational arithmetic, not rounded decimals.S073 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S072 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 with exact rational arithmetic, not rounded decimals.S071 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S070 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 with exact rational arithmetic, not rounded decimals.S069 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S068 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 with exact rational arithmetic, not rounded decimals.S067 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S066 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 with exact rational arithmetic, not rounded decimals.S065 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S064 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 with exact rational arithmetic, not rounded decimals.S063 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S062 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 with exact rational arithmetic, not rounded decimals.S061 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S060 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 with exact rational arithmetic, not rounded decimals.S059 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S058 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 with exact rational arithmetic, not rounded decimals.S057 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S056 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 with exact rational arithmetic, not rounded decimals.S055 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S054 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 with exact rational arithmetic, not rounded decimals.S053 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S052 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 with exact rational arithmetic, not rounded decimals.S051 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S050 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 with exact rational arithmetic, not rounded decimals.S049 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S048 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 with exact rational arithmetic, not rounded decimals.S047 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S046 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 with exact rational arithmetic, not rounded decimals.S045 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S044 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S042 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S040 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S038 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S036 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S034 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S032 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S030 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S028 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S026 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S024 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S022 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S020 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S018 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S016 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S014 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S012 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S010 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product5 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S008 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product4 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S006 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product3 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S004 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product2 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S002 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product1 with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.(fermat_confusion, 97, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 23], [6, 41], [7, 93], [8, 65], [9, 3], [10, 30], [11, 39], [12, 80], [13, 70], [14, 10], [15, 53], [16, 72], [17, 60], [18, 13], [19, 53], [20, 90], [21, 47], [22, 64], [23, 17], [24, 20], [25, 15], [26, 2], [27, 54], [28, 57], [29, 4], [30, 23], [31, 34], [32, 21], [33, 14], [34, 88], [35, 73], [36, 9], [37, 42], [38, 44], [39, 67], [40, 61], [41, 76], [42, 88], [43, 1], [44, 44], [45, 40], [46, 94], [47, 53], [48, 22], [49, 11], [50, 65], [51, 17], [52, 11], [53, 1], [54, 54], [55, 60], [56, 62], [57, 42], [58, 11], [59, 67], [60, 43], [61, 4], [62, 54], [63, 7], [64, 60], [65, 20], [66, 59], [67, 73], [68, 17], [69, 9], [70, 48], [71, 13], [72, 63], [73, 40], [74, 50], [75, 64], [76, 14], [77, 11], [78, 82], [79, 76], [80, 66], [81, 11], [82, 29], [83, 79], [84, 40], [85, 5], [86, 42], [87, 65], [88, 94], [89, 24], [90, 26], [91, 38], [92, 4], [93, 81], [94, 48], [95, 1], [96, 96]], 96, 96, Yes, Wilson:(n-1)!==-1_mod_97, Fermat:a^(97-1)==1_mod_97_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 0a9934517116a1f6aed3acd34d94fd0275b63ae29f200df0da0735cb2da3125bcandidate record SHA-256: 6dfb3b566b1f87c8cb4fe311ca5bd52e25fe71ddd5a1f2f3dd033cf00746952asemantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED