Explore related knowledge and questions →
← Question derivation index · Original question batch
R0054-27716 · q-20261015-advance-20261118-r04-wilson-theorem-0091
Use mode=fermat_confusion and exact integer n=53. 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=53. 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, 53, 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, 53, 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, 53, 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, 53, 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, 53, 5, 24, 120]
After
derived = [2, 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, S010
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 := mul(14, 6)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 = 84
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, 53, 6, 14, 84]
After
derived = [1, 31, 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(31, 7)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 = 217
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, 53, 7, 31, 217]
After
derived = [4, 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, S014
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 := mul(5, 8)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 = 40
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, 53, 8, 5, 40]
After
derived = [0, 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, S016
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 := mul(40, 9)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 = 360
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, 53, 9, 40, 360]
After
derived = [6, 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, S018
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 := mul(42, 10)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 = 420
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, 53, 10, 42, 420]
After
derived = [7, 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, S020
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 := mul(49, 11)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 = 539
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, 53, 11, 49, 539]
After
derived = [10, 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, 53, 12, 9, 108]
After
derived = [2, 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, S024
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 := mul(2, 13)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 = 26
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, 53, 13, 2, 26]
After
derived = [0, 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, S026
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 := mul(26, 14)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 = 364
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, 53, 14, 26, 364]
After
derived = [6, 46, 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(46, 15)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 = 690
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, 53, 15, 46, 690]
After
derived = [13, 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, S030
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 := mul(1, 16)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 = 16
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, 53, 16, 1, 16]
After
derived = [0, 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, S032
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 := mul(16, 17)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 = 272
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, 53, 17, 16, 272]
After
derived = [5, 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, S034
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 := mul(7, 18)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 = 126
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, 53, 18, 7, 126]
After
derived = [2, 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, S036
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 := mul(20, 19)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 = 380
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, 53, 19, 20, 380]
After
derived = [7, 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, S038
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 := mul(9, 20)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 = 180
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, 53, 20, 9, 180]
After
derived = [3, 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, S040
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 := mul(21, 21)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 = 441
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, 53, 21, 21, 441]
After
derived = [8, 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, S042
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 := mul(17, 22)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 = 374
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, 53, 22, 17, 374]
After
derived = [7, 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, S044
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 := mul(3, 23)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 = 69
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, 53, 23, 3, 69]
After
derived = [1, 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, S046
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 := mul(16, 24)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 = 384
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, 53, 24, 16, 384]
After
derived = [7, 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, S048
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 := mul(13, 25)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 = 325
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, 53, 25, 13, 325]
After
derived = [6, 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, S050
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 := mul(7, 26)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 = 182
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, 53, 26, 7, 182]
After
derived = [3, 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, S052
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 := mul(23, 27)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 = 621
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, 53, 27, 23, 621]
After
derived = [11, 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, S054
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 := mul(38, 28)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 = 1064
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, 53, 28, 38, 1064]
After
derived = [20, 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, S056
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 := mul(4, 29)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 = 116
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, 53, 29, 4, 116]
After
derived = [2, 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, S058
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 := mul(10, 30)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 = 300
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, 53, 30, 10, 300]
After
derived = [5, 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, S060
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 := mul(35, 31)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 = 1085
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, 53, 31, 35, 1085]
After
derived = [20, 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, S062
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 := mul(25, 32)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 = 800
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, 53, 32, 25, 800]
After
derived = [15, 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, S064
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 := mul(5, 33)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 = 165
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, 53, 33, 5, 165]
After
derived = [3, 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, S066
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 := mul(6, 34)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 = 204
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, 53, 34, 6, 204]
After
derived = [3, 45, 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(45, 35)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 = 1575
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, 53, 35, 45, 1575]
After
derived = [29, 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, S070
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 := mul(38, 36)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 = 1368
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, 53, 36, 38, 1368]
After
derived = [25, 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, S072
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 := mul(43, 37)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 = 1591
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, 53, 37, 43, 1591]
After
derived = [30, 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, S074
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 := mul(1, 38)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 = 38
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, 53, 38, 1, 38]
After
derived = [0, 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, S076
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 := mul(38, 39)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 = 1482
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, 53, 39, 38, 1482]
After
derived = [27, 51, 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(51, 40)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 = 2040
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, 53, 40, 51, 2040]
After
derived = [38, 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, S080
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 := mul(26, 41)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 = 1066
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, 53, 41, 26, 1066]
After
derived = [20, 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, S082
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 := mul(6, 42)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 = 252
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, 53, 42, 6, 252]
After
derived = [4, 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, S084
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product43 := mul(40, 43)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product43 = 1720
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, 53, 43, 40, 1720]
After
derived = [32, 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, S086
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 := mul(24, 44)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 = 1056
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, 53, 44, 24, 1056]
After
derived = [19, 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, S088
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 := mul(49, 45)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 = 2205
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, 53, 45, 49, 2205]
After
derived = [41, 32, 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(32, 46)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product46 = 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, S091
Before
source roles = [Wilson boundary ORIGINAL prime, 53, 46, 32, 1472]
After
derived = [27, 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, S092
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 := mul(41, 47)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 = 1927
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, 53, 47, 41, 1927]
After
derived = [36, 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, S094
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 := mul(19, 48)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 = 912
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, 53, 48, 19, 912]
After
derived = [17, 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, S096
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product49 := mul(11, 49)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product49 = 539
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, 53, 49, 11, 539]
After
derived = [10, 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, S098
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product50 := mul(9, 50)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product50 = 450
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, 53, 50, 9, 450]
After
derived = [8, 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, S100
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product51 := mul(26, 51)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product51 = 1326
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, 53, 51, 26, 1326]
After
derived = [25, 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, S102
Before
Wilson boundary ORIGINAL prime ORIGINAL running factorial product52 := mul(1, 52)
After
Wilson boundary ORIGINAL prime ORIGINAL running factorial product52 = 52
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, 53, 52, 1, 52]
After
derived = [0, 52, 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
source roles = [53, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 14], [6, 31], [7, 5], [8, 40], [9, 42], [10, 49], [11, 9], [12, 2], [13, 26], [14, 46], [15, 1], [16, 16], [17, 7], [18, 20], [19, 9], [20, 21], [21, 17], [22, 3], [23, 16], [24, 13], [25, 7], [26, 23], [27, 38], [28, 4], [29, 10], [30, 35], [31, 25], [32, 5], [33, 6], [34, 45], [35, 38], [36, 43], [37, 1], [38, 38], [39, 51], [40, 26], [41, 6], [42, 40], [43, 24], [44, 49], [45, 32], [46, 41], [47, 19], [48, 11], [49, 9], [50, 26], [51, 1], [52, 52]], 52]
After
derived = [52, 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, S105
S106 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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, 53, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 14], [6, 31], [7, 5], [8, 40], [9, 42], [10, 49], [11, 9], [12, 2], [13, 26], [14, 46], [15, 1], [16, 16], [17, 7], [18, 20], [19, 9], [20, 21], [21, 17], [22, 3], [23, 16], [24, 13], [25, 7], [26, 23], [27, 38], [28, 4], [29, 10], [30, 35], [31, 25], [32, 5], [33, 6], [34, 45], [35, 38], [36, 43], [37, 1], [38, 38], [39, 51], [40, 26], [41, 6], [42, 40], [43, 24], [44, 49], [45, 32], [46, 41], [47, 19], [48, 11], [49, 9], [50, 26], [51, 1], [52, 52]], 52, 52, Yes, Wilson:(n-1)!==-1_mod_53, Fermat:a^(53-1)==1_mod_53_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: 7f3c728274649d991858070e3bcd2814e163ac096df2f33dbf3f463625272da1candidate record SHA-256: e1e17aab842d0b2d744ecf4cc4f1eb2348cf005fe7518b6bb7061fa1c86c64fcsemantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED