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R0054-27788 · q-20261015-advance-20261118-r04-wilson-theorem-0038
Use prime modulus p=41. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).
G001 — Use prime modulus p=41. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).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-prime-inverse-pairing]
After
derived = [[p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence]]
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
source roles = [41, 1]
After
derived = [1, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S001
Before
source roles = [41, 2]
After
derived = [21, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S002
Before
source roles = [41, 3]
After
derived = [14, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S003
Before
source roles = [41, 4]
After
derived = [31, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S004
Before
source roles = [41, 5]
After
derived = [33, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S005
Before
source roles = [41, 6]
After
derived = [7, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S006
Before
source roles = [41, 7]
After
derived = [6, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S007
Before
source roles = [41, 8]
After
derived = [36, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S008
Before
source roles = [41, 9]
After
derived = [32, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S009
Before
source roles = [41, 10]
After
derived = [37, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S010
Before
source roles = [41, 11]
After
derived = [15, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S011
Before
source roles = [41, 12]
After
derived = [24, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S012
Before
source roles = [41, 13]
After
derived = [19, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S013
Before
source roles = [41, 14]
After
derived = [3, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S014
Before
source roles = [41, 15]
After
derived = [11, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S015
Before
source roles = [41, 16]
After
derived = [18, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S016
Before
source roles = [41, 17]
After
derived = [29, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S017
Before
source roles = [41, 18]
After
derived = [16, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S018
Before
source roles = [41, 19]
After
derived = [13, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S019
Before
source roles = [41, 20]
After
derived = [39, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S020
Before
source roles = [41, 21]
After
derived = [2, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S021
Before
source roles = [41, 22]
After
derived = [28, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S022
Before
source roles = [41, 23]
After
derived = [25, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S023
Before
source roles = [41, 24]
After
derived = [12, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S024
Before
source roles = [41, 25]
After
derived = [23, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S025
Before
source roles = [41, 26]
After
derived = [30, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S026
Before
source roles = [41, 27]
After
derived = [38, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S027
Before
source roles = [41, 28]
After
derived = [22, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S028
Before
source roles = [41, 29]
After
derived = [17, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S029
Before
source roles = [41, 30]
After
derived = [26, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S030
Before
source roles = [41, 31]
After
derived = [4, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S031
Before
source roles = [41, 32]
After
derived = [9, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S032
Before
source roles = [41, 33]
After
derived = [5, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S033
Before
source roles = [41, 34]
After
derived = [35, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S034
Before
source roles = [41, 35]
After
derived = [34, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S035
Before
source roles = [41, 36]
After
derived = [8, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S036
Before
source roles = [41, 37]
After
derived = [10, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S037
Before
source roles = [41, 38]
After
derived = [27, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S038
Before
source roles = [41, 39]
After
derived = [20, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S039
Before
source roles = [41, 40]
After
derived = [40, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S040
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 := mul(1, 1)
After
Wilson ORIGINAL prime factorial 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, S041
Before
source roles = [Wilson ORIGINAL prime factorial, 41, 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, S042
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 := mul(1, 2)
After
Wilson ORIGINAL prime factorial 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, S043
Before
source roles = [Wilson ORIGINAL prime factorial, 41, 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, S044
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 := mul(2, 3)
After
Wilson ORIGINAL prime factorial 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, S045
Before
source roles = [Wilson ORIGINAL prime factorial, 41, 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, S046
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 := mul(6, 4)
After
Wilson ORIGINAL prime factorial 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, S047
Before
source roles = [Wilson ORIGINAL prime factorial, 41, 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, S048
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 := mul(24, 5)
After
Wilson ORIGINAL prime factorial 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, S049
Before
source roles = [Wilson ORIGINAL prime factorial, 41, 5, 24, 120]
After
derived = [2, 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, S050
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(38, 6)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 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, S051
Before
source roles = [Wilson ORIGINAL prime factorial, 41, 6, 38, 228]
After
derived = [5, 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 ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(23, 7)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 161
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 ORIGINAL prime factorial, 41, 7, 23, 161]
After
derived = [3, 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 ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(38, 8)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 304
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 ORIGINAL prime factorial, 41, 8, 38, 304]
After
derived = [7, 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, S056
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(17, 9)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 153
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 ORIGINAL prime factorial, 41, 9, 17, 153]
After
derived = [3, 30, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S058
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(30, 10)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 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 ORIGINAL prime factorial, 41, 10, 30, 300]
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, S060
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(13, 11)
After
Wilson ORIGINAL prime factorial 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, S061
Before
source roles = [Wilson ORIGINAL prime factorial, 41, 11, 13, 143]
After
derived = [3, 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, S062
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(20, 12)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 240
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 ORIGINAL prime factorial, 41, 12, 20, 240]
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, S064
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(35, 13)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 455
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 ORIGINAL prime factorial, 41, 13, 35, 455]
After
derived = [11, 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, S066
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(4, 14)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 56
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 ORIGINAL prime factorial, 41, 14, 4, 56]
After
derived = [1, 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, S068
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(15, 15)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 225
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 ORIGINAL prime factorial, 41, 15, 15, 225]
After
derived = [5, 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, S070
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(20, 16)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 320
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 ORIGINAL prime factorial, 41, 16, 20, 320]
After
derived = [7, 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, S072
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(33, 17)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 561
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 ORIGINAL prime factorial, 41, 17, 33, 561]
After
derived = [13, 28, 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 ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(28, 18)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 504
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 ORIGINAL prime factorial, 41, 18, 28, 504]
After
derived = [12, 12, 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 ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(12, 19)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 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, S077
Before
source roles = [Wilson ORIGINAL prime factorial, 41, 19, 12, 228]
After
derived = [5, 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, S078
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(23, 20)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 460
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 ORIGINAL prime factorial, 41, 20, 23, 460]
After
derived = [11, 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, S080
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 := mul(9, 21)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 189
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 ORIGINAL prime factorial, 41, 21, 9, 189]
After
derived = [4, 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, S082
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(25, 22)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 550
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S083
Before
source roles = [Wilson ORIGINAL prime factorial, 41, 22, 25, 550]
After
derived = [13, 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, S084
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(17, 23)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 391
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 ORIGINAL prime factorial, 41, 23, 17, 391]
After
derived = [9, 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, S086
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 := mul(22, 24)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 528
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 ORIGINAL prime factorial, 41, 24, 22, 528]
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, S088
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 := mul(36, 25)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 900
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 ORIGINAL prime factorial, 41, 25, 36, 900]
After
derived = [21, 39, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S090
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 := mul(39, 26)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 1014
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 ORIGINAL prime factorial, 41, 26, 39, 1014]
After
derived = [24, 30, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S092
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(30, 27)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 810
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 ORIGINAL prime factorial, 41, 27, 30, 810]
After
derived = [19, 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, S094
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(31, 28)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 868
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 ORIGINAL prime factorial, 41, 28, 31, 868]
After
derived = [21, 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, S096
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(7, 29)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 203
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 ORIGINAL prime factorial, 41, 29, 7, 203]
After
derived = [4, 39, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S098
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(39, 30)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 1170
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 ORIGINAL prime factorial, 41, 30, 39, 1170]
After
derived = [28, 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, S100
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 := mul(22, 31)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 = 682
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 ORIGINAL prime factorial, 41, 31, 22, 682]
After
derived = [16, 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, S102
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 := mul(26, 32)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 = 832
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 ORIGINAL prime factorial, 41, 32, 26, 832]
After
derived = [20, 12, 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 ORIGINAL prime factorial ORIGINAL running factorial product33 := mul(12, 33)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 = 396
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 ORIGINAL prime factorial, 41, 33, 12, 396]
After
derived = [9, 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, S106
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 := mul(27, 34)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 = 918
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 ORIGINAL prime factorial, 41, 34, 27, 918]
After
derived = [22, 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, S108
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 := mul(16, 35)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 = 560
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 ORIGINAL prime factorial, 41, 35, 16, 560]
After
derived = [13, 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, S110
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 := mul(27, 36)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 = 972
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 ORIGINAL prime factorial, 41, 36, 27, 972]
After
derived = [23, 29, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S112
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 := mul(29, 37)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 = 1073
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 ORIGINAL prime factorial, 41, 37, 29, 1073]
After
derived = [26, 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, S114
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 := mul(7, 38)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 = 266
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 ORIGINAL prime factorial, 41, 38, 7, 266]
After
derived = [6, 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, S116
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 := mul(20, 39)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 = 780
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 ORIGINAL prime factorial, 41, 39, 20, 780]
After
derived = [19, 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, S118
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 := mul(1, 40)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 = 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, S119
Before
source roles = [Wilson ORIGINAL prime factorial, 41, 40, 1, 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, S120
Before
source roles = [41, [[1, 1], [2, 21], [3, 14], [4, 31], [5, 33], [6, 7], [7, 6], [8, 36], [9, 32], [10, 37], [11, 15], [12, 24], [13, 19], [14, 3], [15, 11], [16, 18], [17, 29], [18, 16], [19, 13], [20, 39], [21, 2], [22, 28], [23, 25], [24, 12], [25, 23], [26, 30], [27, 38], [28, 22], [29, 17], [30, 26], [31, 4], [32, 9], [33, 5], [34, 35], [35, 34], [36, 8], [37, 10], [38, 27], [39, 20], [40, 40]]]
After
derived = [[1, 40], [[2, 21, 1], [3, 14, 1], [4, 31, 1], [5, 33, 1], [6, 7, 1], [8, 36, 1], [9, 32, 1], [10, 37, 1], [11, 15, 1], [12, 24, 1], [13, 19, 1], [16, 18, 1], [17, 29, 1], [20, 39, 1], [22, 28, 1], [23, 25, 1], [26, 30, 1], [27, 38, 1], [34, 35, 1]], 40, Yes]
Conditions: none
Nonself inverse pairs multiply to1.A self-inverse satisfies(r-1)(r+1)=0 modulo a prime,so is±1.For p>2 their product is-1;at p2 only the single1=-1 remains.This general teaching argument does not turn the finite table/program into a Lean proof.
Rule: Wilson_GENERAL_inverse_pairing_and_p2_exception · Uses: G001, S121
S122 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 ORIGINAL prime factorial ORIGINAL running factorial product40 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 ORIGINAL prime factorial ORIGINAL running factorial product39 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 ORIGINAL prime factorial ORIGINAL running factorial product38 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 ORIGINAL prime factorial ORIGINAL running factorial product37 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 ORIGINAL prime factorial ORIGINAL running factorial product36 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 ORIGINAL prime factorial ORIGINAL running factorial product35 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 ORIGINAL prime factorial ORIGINAL running factorial product34 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 ORIGINAL prime factorial ORIGINAL running factorial product33 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 ORIGINAL prime factorial ORIGINAL running factorial product32 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 ORIGINAL prime factorial ORIGINAL running factorial product31 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 ORIGINAL prime factorial ORIGINAL running factorial product30 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 ORIGINAL prime factorial ORIGINAL running factorial product29 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 ORIGINAL prime factorial ORIGINAL running factorial product28 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 ORIGINAL prime factorial ORIGINAL running factorial product27 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 ORIGINAL prime factorial ORIGINAL running factorial product26 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 ORIGINAL prime factorial ORIGINAL running factorial product25 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 ORIGINAL prime factorial ORIGINAL running factorial product24 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 ORIGINAL prime factorial ORIGINAL running factorial product23 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 ORIGINAL prime factorial ORIGINAL running factorial product22 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 ORIGINAL prime factorial ORIGINAL running factorial product21 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 ORIGINAL prime factorial ORIGINAL running factorial product20 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 ORIGINAL prime factorial ORIGINAL running factorial product19 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 ORIGINAL prime factorial ORIGINAL running factorial product18 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 ORIGINAL prime factorial ORIGINAL running factorial product17 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 ORIGINAL prime factorial ORIGINAL running factorial product16 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 ORIGINAL prime factorial ORIGINAL running factorial product15 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 ORIGINAL prime factorial ORIGINAL running factorial product14 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 ORIGINAL prime factorial ORIGINAL running factorial product13 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 ORIGINAL prime factorial ORIGINAL running factorial product12 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 ORIGINAL prime factorial ORIGINAL running factorial product11 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 ORIGINAL prime factorial ORIGINAL running factorial product10 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 ORIGINAL prime factorial ORIGINAL running factorial product9 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 ORIGINAL prime factorial ORIGINAL running factorial product8 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 ORIGINAL prime factorial ORIGINAL running factorial product7 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 ORIGINAL prime factorial ORIGINAL running factorial product6 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 ORIGINAL prime factorial ORIGINAL running factorial product5 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 ORIGINAL prime factorial ORIGINAL running factorial product4 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 ORIGINAL prime factorial ORIGINAL running factorial product3 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 ORIGINAL prime factorial ORIGINAL running factorial product2 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 ORIGINAL prime factorial ORIGINAL running factorial product1 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.(41, [[1, 1], [2, 21], [3, 14], [4, 31], [5, 33], [6, 7], [7, 6], [8, 36], [9, 32], [10, 37], [11, 15], [12, 24], [13, 19], [14, 3], [15, 11], [16, 18], [17, 29], [18, 16], [19, 13], [20, 39], [21, 2], [22, 28], [23, 25], [24, 12], [25, 23], [26, 30], [27, 38], [28, 22], [29, 17], [30, 26], [31, 4], [32, 9], [33, 5], [34, 35], [35, 34], [36, 8], [37, 10], [38, 27], [39, 20], [40, 40]], [1, 40], [[2, 21, 1], [3, 14, 1], [4, 31, 1], [5, 33, 1], [6, 7, 1], [8, 36, 1], [9, 32, 1], [10, 37, 1], [11, 15, 1], [12, 24, 1], [13, 19, 1], [16, 18, 1], [17, 29, 1], [20, 39, 1], [22, 28, 1], [23, 25, 1], [26, 30, 1], [27, 38, 1], [34, 35, 1]], self_inverse_residues_1_and_p_minus_1, 40, Yes)
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: acf49f42709a1f3f24f6a05a8aa64c782b9730f72ed6bd04843d53b5f0008ee3candidate record SHA-256: b61efa4ff7e833c3a8e0c98fb51098ae9da30bbe77161f05b9d7d22807a89afesemantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED