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R0054-27719 · q-20261015-advance-20261118-r04-wilson-theorem-0094
Use mode=fermat_confusion and exact integer n=67. 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=67. 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, 67, 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, 67, 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, 67, 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, 67, 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, 67, 5, 24, 120]
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, S010
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 := mul(53, 6)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 = 318
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, 67, 6, 53, 318]
After
derived = [4, 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, S012
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 := mul(50, 7)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 = 350
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, 67, 7, 50, 350]
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, S014
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 := mul(15, 8)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 = 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, S015
Before
source roles = [Wilson boundary ORIGINAL prime, 67, 8, 15, 120]
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, S016
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 := mul(53, 9)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 = 477
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, 67, 9, 53, 477]
After
derived = [7, 8, 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(8, 10)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 = 80
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, 67, 10, 8, 80]
After
derived = [1, 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, S020
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 := mul(13, 11)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 = 143
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, 67, 11, 13, 143]
After
derived = [2, 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, S022
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 := mul(9, 12)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 = 108
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, 67, 12, 9, 108]
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, S024
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 := mul(41, 13)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 = 533
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, 67, 13, 41, 533]
After
derived = [7, 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, S026
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 := mul(64, 14)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 = 896
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, 67, 14, 64, 896]
After
derived = [13, 25, 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(25, 15)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 = 375
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, 67, 15, 25, 375]
After
derived = [5, 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, S030
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 := mul(40, 16)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 = 640
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, 67, 16, 40, 640]
After
derived = [9, 37, 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(37, 17)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 = 629
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, 67, 17, 37, 629]
After
derived = [9, 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, S034
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 := mul(26, 18)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 = 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, S035
Before
source roles = [Wilson boundary ORIGINAL prime, 67, 18, 26, 468]
After
derived = [6, 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, S036
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 := mul(66, 19)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 = 1254
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, 67, 19, 66, 1254]
After
derived = [18, 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, S038
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 := mul(48, 20)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 = 960
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, 67, 20, 48, 960]
After
derived = [14, 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, S040
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 := mul(22, 21)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 = 462
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, 67, 21, 22, 462]
After
derived = [6, 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, S042
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 := mul(60, 22)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 = 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, S043
Before
source roles = [Wilson boundary ORIGINAL prime, 67, 22, 60, 1320]
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, S044
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 := mul(47, 23)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 = 1081
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, 67, 23, 47, 1081]
After
derived = [16, 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, S046
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 := mul(9, 24)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 = 216
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, 67, 24, 9, 216]
After
derived = [3, 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, S048
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 := mul(15, 25)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 = 375
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, 67, 25, 15, 375]
After
derived = [5, 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, S050
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 := mul(40, 26)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 = 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, S051
Before
source roles = [Wilson boundary ORIGINAL prime, 67, 26, 40, 1040]
After
derived = [15, 35, 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(35, 27)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 = 945
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, 67, 27, 35, 945]
After
derived = [14, 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, S054
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 := mul(7, 28)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 = 196
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, 67, 28, 7, 196]
After
derived = [2, 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, S056
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 := mul(62, 29)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 = 1798
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, 67, 29, 62, 1798]
After
derived = [26, 56, 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(56, 30)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 = 1680
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, 67, 30, 56, 1680]
After
derived = [25, 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, S060
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 := mul(5, 31)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 = 155
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, 67, 31, 5, 155]
After
derived = [2, 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, S062
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 := mul(21, 32)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 = 672
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, 67, 32, 21, 672]
After
derived = [10, 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, S064
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 := mul(2, 33)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 = 66
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, 67, 33, 2, 66]
After
derived = [0, 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, S066
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 := mul(66, 34)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 = 2244
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, 67, 34, 66, 2244]
After
derived = [33, 33, 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(33, 35)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 = 1155
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, 67, 35, 33, 1155]
After
derived = [17, 16, 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(16, 36)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 = 576
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, 67, 36, 16, 576]
After
derived = [8, 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, S072
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 := mul(40, 37)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 = 1480
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, 67, 37, 40, 1480]
After
derived = [22, 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, S074
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 := mul(6, 38)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 = 228
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, 67, 38, 6, 228]
After
derived = [3, 27, 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(27, 39)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 = 1053
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, 67, 39, 27, 1053]
After
derived = [15, 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, S078
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 := mul(48, 40)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 = 1920
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, 67, 40, 48, 1920]
After
derived = [28, 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, S080
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 := mul(44, 41)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 = 1804
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, 67, 41, 44, 1804]
After
derived = [26, 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, S082
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 := mul(62, 42)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 = 2604
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, 67, 42, 62, 2604]
After
derived = [38, 58, 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(58, 43)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product43 = 2494
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, 67, 43, 58, 2494]
After
derived = [37, 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, S086
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 := mul(15, 44)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 = 660
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, 67, 44, 15, 660]
After
derived = [9, 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, S088
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 := mul(57, 45)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 = 2565
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, 67, 45, 57, 2565]
After
derived = [38, 19, 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(19, 46)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product46 = 874
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, 67, 46, 19, 874]
After
derived = [13, 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, S092
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 := mul(3, 47)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 = 141
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, 67, 47, 3, 141]
After
derived = [2, 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, S094
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 := mul(7, 48)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 = 336
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, 67, 48, 7, 336]
After
derived = [5, 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, S096
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product49 := mul(1, 49)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product49 = 49
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, 67, 49, 1, 49]
After
derived = [0, 49, 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(49, 50)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product50 = 2450
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, 67, 50, 49, 2450]
After
derived = [36, 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, S100
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product51 := mul(38, 51)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product51 = 1938
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, 67, 51, 38, 1938]
After
derived = [28, 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, S102
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product52 := mul(62, 52)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product52 = 3224
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, 67, 52, 62, 3224]
After
derived = [48, 8, 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(8, 53)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product53 = 424
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, 67, 53, 8, 424]
After
derived = [6, 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, S106
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product54 := mul(22, 54)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product54 = 1188
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, 67, 54, 22, 1188]
After
derived = [17, 49, 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(49, 55)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product55 = 2695
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, 67, 55, 49, 2695]
After
derived = [40, 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, S110
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product56 := mul(15, 56)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product56 = 840
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, 67, 56, 15, 840]
After
derived = [12, 36, 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(36, 57)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product57 = 2052
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, 67, 57, 36, 2052]
After
derived = [30, 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, 67, 58, 42, 2436]
After
derived = [36, 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, S116
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product59 := mul(24, 59)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product59 = 1416
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, 67, 59, 24, 1416]
After
derived = [21, 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, S118
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product60 := mul(9, 60)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product60 = 540
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, 67, 60, 9, 540]
After
derived = [8, 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, S120
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product61 := mul(4, 61)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product61 = 244
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, 67, 61, 4, 244]
After
derived = [3, 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, S122
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product62 := mul(43, 62)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product62 = 2666
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, 67, 62, 43, 2666]
After
derived = [39, 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, S124
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product63 := mul(53, 63)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product63 = 3339
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, 67, 63, 53, 3339]
After
derived = [49, 56, 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(56, 64)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product64 = 3584
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, 67, 64, 56, 3584]
After
derived = [53, 33, 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(33, 65)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product65 = 2145
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, 67, 65, 33, 2145]
After
derived = [32, 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, S130
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product66 := mul(1, 66)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product66 = 66
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, 67, 66, 1, 66]
After
derived = [0, 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, S132
Before
source roles = [67, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 53], [6, 50], [7, 15], [8, 53], [9, 8], [10, 13], [11, 9], [12, 41], [13, 64], [14, 25], [15, 40], [16, 37], [17, 26], [18, 66], [19, 48], [20, 22], [21, 60], [22, 47], [23, 9], [24, 15], [25, 40], [26, 35], [27, 7], [28, 62], [29, 56], [30, 5], [31, 21], [32, 2], [33, 66], [34, 33], [35, 16], [36, 40], [37, 6], [38, 27], [39, 48], [40, 44], [41, 62], [42, 58], [43, 15], [44, 57], [45, 19], [46, 3], [47, 7], [48, 1], [49, 49], [50, 38], [51, 62], [52, 8], [53, 22], [54, 49], [55, 15], [56, 36], [57, 42], [58, 24], [59, 9], [60, 4], [61, 43], [62, 53], [63, 56], [64, 33], [65, 1], [66, 66]], 66]
After
derived = [66, 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, S133
S134 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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, 67, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 53], [6, 50], [7, 15], [8, 53], [9, 8], [10, 13], [11, 9], [12, 41], [13, 64], [14, 25], [15, 40], [16, 37], [17, 26], [18, 66], [19, 48], [20, 22], [21, 60], [22, 47], [23, 9], [24, 15], [25, 40], [26, 35], [27, 7], [28, 62], [29, 56], [30, 5], [31, 21], [32, 2], [33, 66], [34, 33], [35, 16], [36, 40], [37, 6], [38, 27], [39, 48], [40, 44], [41, 62], [42, 58], [43, 15], [44, 57], [45, 19], [46, 3], [47, 7], [48, 1], [49, 49], [50, 38], [51, 62], [52, 8], [53, 22], [54, 49], [55, 15], [56, 36], [57, 42], [58, 24], [59, 9], [60, 4], [61, 43], [62, 53], [63, 56], [64, 33], [65, 1], [66, 66]], 66, 66, Yes, Wilson:(n-1)!==-1_mod_67, Fermat:a^(67-1)==1_mod_67_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: 88c99669ed29f8922e02d80ec8c9e9cd40fc74d723a1675c75e3db0b148a2279candidate record SHA-256: 1abb11f945c24f8db354d16f527ee6b1e9fa30a0af074cf0022e29e59ad113d9semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED