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R0054-27784 · q-20261015-advance-20261118-r04-wilson-theorem-0034
Use prime modulus p=23. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).
G001 — Use prime modulus p=23. 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 = [23, 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 = [23, 2]
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, S002
Before
source roles = [23, 3]
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, S003
Before
source roles = [23, 4]
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, S004
Before
source roles = [23, 5]
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, S005
Before
source roles = [23, 6]
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, S006
Before
source roles = [23, 7]
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, S007
Before
source roles = [23, 8]
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, S008
Before
source roles = [23, 9]
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, S009
Before
source roles = [23, 10]
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, S010
Before
source roles = [23, 11]
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, S011
Before
source roles = [23, 12]
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, S012
Before
source roles = [23, 13]
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, S013
Before
source roles = [23, 14]
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, S014
Before
source roles = [23, 15]
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, S015
Before
source roles = [23, 16]
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, S016
Before
source roles = [23, 17]
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, S017
Before
source roles = [23, 18]
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, S018
Before
source roles = [23, 19]
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, S019
Before
source roles = [23, 20]
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, S020
Before
source roles = [23, 21]
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, S021
Before
source roles = [23, 22]
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, S022
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, S023
Before
source roles = [Wilson ORIGINAL prime factorial, 23, 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, S024
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, S025
Before
source roles = [Wilson ORIGINAL prime factorial, 23, 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, S026
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, S027
Before
source roles = [Wilson ORIGINAL prime factorial, 23, 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, S028
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, S029
Before
source roles = [Wilson ORIGINAL prime factorial, 23, 4, 6, 24]
After
derived = [1, 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 ORIGINAL prime factorial ORIGINAL running factorial product5 := mul(1, 5)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 = 5
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 ORIGINAL prime factorial, 23, 5, 1, 5]
After
derived = [0, 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, S032
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(5, 6)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 30
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
source roles = [Wilson ORIGINAL prime factorial, 23, 6, 5, 30]
After
derived = [1, 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 ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(7, 7)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 49
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
source roles = [Wilson ORIGINAL prime factorial, 23, 7, 7, 49]
After
derived = [2, 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, S036
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(3, 8)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 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, S037
Before
source roles = [Wilson ORIGINAL prime factorial, 23, 8, 3, 24]
After
derived = [1, 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, S038
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(1, 9)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 9
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 ORIGINAL prime factorial, 23, 9, 1, 9]
After
derived = [0, 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, S040
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(9, 10)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 90
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, 23, 10, 9, 90]
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, S042
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(21, 11)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 231
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, 23, 11, 21, 231]
After
derived = [10, 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, S044
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(1, 12)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 12
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, 23, 12, 1, 12]
After
derived = [0, 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, S046
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(12, 13)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 156
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, 23, 13, 12, 156]
After
derived = [6, 18, 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 product14 := mul(18, 14)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 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, S049
Before
source roles = [Wilson ORIGINAL prime factorial, 23, 14, 18, 252]
After
derived = [10, 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, S050
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(22, 15)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 330
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
source roles = [Wilson ORIGINAL prime factorial, 23, 15, 22, 330]
After
derived = [14, 8, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S052
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(8, 16)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 128
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, 23, 16, 8, 128]
After
derived = [5, 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, S054
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(13, 17)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 221
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, 23, 17, 13, 221]
After
derived = [9, 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, S056
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(14, 18)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 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, S057
Before
source roles = [Wilson ORIGINAL prime factorial, 23, 18, 14, 252]
After
derived = [10, 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, S058
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(22, 19)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 418
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, 23, 19, 22, 418]
After
derived = [18, 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, S060
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(4, 20)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 80
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
source roles = [Wilson ORIGINAL prime factorial, 23, 20, 4, 80]
After
derived = [3, 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, S062
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 := mul(11, 21)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 231
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, 23, 21, 11, 231]
After
derived = [10, 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, S064
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(1, 22)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 22
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, 23, 22, 1, 22]
After
derived = [0, 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, S066
Before
source roles = [23, [[1, 1], [2, 12], [3, 8], [4, 6], [5, 14], [6, 4], [7, 10], [8, 3], [9, 18], [10, 7], [11, 21], [12, 2], [13, 16], [14, 5], [15, 20], [16, 13], [17, 19], [18, 9], [19, 17], [20, 15], [21, 11], [22, 22]]]
After
derived = [[1, 22], [[2, 12, 1], [3, 8, 1], [4, 6, 1], [5, 14, 1], [7, 10, 1], [9, 18, 1], [11, 21, 1], [13, 16, 1], [15, 20, 1], [17, 19, 1]], 22, 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, S067
S068 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.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 product22 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 product21 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 product20 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 product19 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 product18 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 product17 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 product16 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 product15 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 product14 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 product13 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 product12 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 product11 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 product10 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 ORIGINAL prime factorial ORIGINAL running factorial product9 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 ORIGINAL prime factorial ORIGINAL running factorial product8 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 ORIGINAL prime factorial ORIGINAL running factorial product7 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 ORIGINAL prime factorial ORIGINAL running factorial product6 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 ORIGINAL prime factorial ORIGINAL running factorial product5 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 ORIGINAL prime factorial ORIGINAL running factorial product4 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 ORIGINAL prime factorial ORIGINAL running factorial product3 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 ORIGINAL prime factorial ORIGINAL running factorial product2 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 ORIGINAL prime factorial ORIGINAL running factorial product1 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 — 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.(23, [[1, 1], [2, 12], [3, 8], [4, 6], [5, 14], [6, 4], [7, 10], [8, 3], [9, 18], [10, 7], [11, 21], [12, 2], [13, 16], [14, 5], [15, 20], [16, 13], [17, 19], [18, 9], [19, 17], [20, 15], [21, 11], [22, 22]], [1, 22], [[2, 12, 1], [3, 8, 1], [4, 6, 1], [5, 14, 1], [7, 10, 1], [9, 18, 1], [11, 21, 1], [13, 16, 1], [15, 20, 1], [17, 19, 1]], self_inverse_residues_1_and_p_minus_1, 22, 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: fe81d427df92e7051986c0117c3fbb12a6726f8a5c8161fbcfe49d74906710facandidate record SHA-256: 69480c4bedae62840f6879503687135e4bd3768c7877a24afac34749c4fe07dasemantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED