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Question 27787: All Nonzero Prime Residues Pair with Inverses, with a Special p=2 Case

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0054-27787 · q-20261015-advance-20261118-r04-wilson-theorem-0037

Original question and givens

Use prime modulus p=37. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).

  • G001 — Use prime modulus p=37. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).

Definition and scope

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.

Step-by-step derivation

S001 · implies

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.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [37, 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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S001

S003 · implies

Before

source roles = [37, 2]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S002

S004 · implies

Before

source roles = [37, 3]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S003

S005 · implies

Before

source roles = [37, 4]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S004

S006 · implies

Before

source roles = [37, 5]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S005

S007 · implies

Before

source roles = [37, 6]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S006

S008 · implies

Before

source roles = [37, 7]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S007

S009 · implies

Before

source roles = [37, 8]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S008

S010 · implies

Before

source roles = [37, 9]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S009

S011 · implies

Before

source roles = [37, 10]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S010

S012 · implies

Before

source roles = [37, 11]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S011

S013 · implies

Before

source roles = [37, 12]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S012

S014 · implies

Before

source roles = [37, 13]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S013

S015 · implies

Before

source roles = [37, 14]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S014

S016 · implies

Before

source roles = [37, 15]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S015

S017 · implies

Before

source roles = [37, 16]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S016

S018 · implies

Before

source roles = [37, 17]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S017

S019 · implies

Before

source roles = [37, 18]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S018

S020 · implies

Before

source roles = [37, 19]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S019

S021 · implies

Before

source roles = [37, 20]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S020

S022 · implies

Before

source roles = [37, 21]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S021

S023 · implies

Before

source roles = [37, 22]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S022

S024 · implies

Before

source roles = [37, 23]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S023

S025 · implies

Before

source roles = [37, 24]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S024

S026 · implies

Before

source roles = [37, 25]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S025

S027 · implies

Before

source roles = [37, 26]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S026

S028 · implies

Before

source roles = [37, 27]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S027

S029 · implies

Before

source roles = [37, 28]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S028

S030 · implies

Before

source roles = [37, 29]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S029

S031 · implies

Before

source roles = [37, 30]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S030

S032 · implies

Before

source roles = [37, 31]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S031

S033 · implies

Before

source roles = [37, 32]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S032

S034 · implies

Before

source roles = [37, 33]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S033

S035 · implies

Before

source roles = [37, 34]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S034

S036 · implies

Before

source roles = [37, 35]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S035

S037 · implies

Before

source roles = [37, 36]

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.

Step identifiers and dependencies

Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S036

S038 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S038

S040 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S040

S042 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S042

S044 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S044

S046 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 5, 24, 120]

After

derived = [3, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S046

S048 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(9, 6)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 54

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 6, 9, 54]

After

derived = [1, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S048

S050 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(17, 7)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 119

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 7, 17, 119]

After

derived = [3, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S050

S052 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(8, 8)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 64

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 8, 8, 64]

After

derived = [1, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S052

S054 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(27, 9)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 243

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 9, 27, 243]

After

derived = [6, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S054

S056 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(21, 10)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 210

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 10, 21, 210]

After

derived = [5, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S056

S058 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(25, 11)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 275

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 11, 25, 275]

After

derived = [7, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S058

S060 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(16, 12)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 192

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 12, 16, 192]

After

derived = [5, 7, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S060

S062 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(7, 13)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 91

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 13, 7, 91]

After

derived = [2, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S062

S064 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(17, 14)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 238

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S063

S065 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 14, 17, 238]

After

derived = [6, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S064

S066 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(16, 15)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S065

S067 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 15, 16, 240]

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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S066

S068 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(18, 16)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 288

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 16, 18, 288]

After

derived = [7, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S068

S070 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(29, 17)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 493

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S069

S071 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 17, 29, 493]

After

derived = [13, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S070

S072 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(12, 18)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 216

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S071

S073 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 18, 12, 216]

After

derived = [5, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S072

S074 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(31, 19)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 589

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S073

S075 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 19, 31, 589]

After

derived = [15, 34, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S074

S076 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(34, 20)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 680

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S075

S077 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 20, 34, 680]

After

derived = [18, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S076

S078 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 := mul(14, 21)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 294

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S077

S079 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 21, 14, 294]

After

derived = [7, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S078

S080 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(35, 22)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 770

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S079

S081 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 22, 35, 770]

After

derived = [20, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S080

S082 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(30, 23)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 690

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S081

S083 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 23, 30, 690]

After

derived = [18, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S082

S084 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 := mul(24, 24)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 576

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S083

S085 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 24, 24, 576]

After

derived = [15, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S084

S086 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 := mul(21, 25)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 525

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S085

S087 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 25, 21, 525]

After

derived = [14, 7, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S086

S088 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 := mul(7, 26)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 182

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S087

S089 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 26, 7, 182]

After

derived = [4, 34, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S088

S090 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(34, 27)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S089

S091 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 27, 34, 918]

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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S090

S092 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(30, 28)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 840

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S091

S093 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 28, 30, 840]

After

derived = [22, 26, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S092

S094 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(26, 29)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 754

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S093

S095 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 29, 26, 754]

After

derived = [20, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S094

S096 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(14, 30)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 420

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S095

S097 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 30, 14, 420]

After

derived = [11, 13, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S096

S098 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 := mul(13, 31)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 = 403

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S097

S099 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 31, 13, 403]

After

derived = [10, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S098

S100 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 := mul(33, 32)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 = 1056

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S099

S101 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 32, 33, 1056]

After

derived = [28, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S100

S102 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 := mul(20, 33)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 = 660

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S101

S103 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 33, 20, 660]

After

derived = [17, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S102

S104 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 := mul(31, 34)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 = 1054

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S103

S105 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 34, 31, 1054]

After

derived = [28, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S104

S106 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 := mul(18, 35)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 = 630

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S105

S107 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 35, 18, 630]

After

derived = [17, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S106

S108 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 := mul(1, 36)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 = 36

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S107

S109 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 37, 36, 1, 36]

After

derived = [0, 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.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S108

S110 · implies

Before

source roles = [37, [[1, 1], [2, 19], [3, 25], [4, 28], [5, 15], [6, 31], [7, 16], [8, 14], [9, 33], [10, 26], [11, 27], [12, 34], [13, 20], [14, 8], [15, 5], [16, 7], [17, 24], [18, 35], [19, 2], [20, 13], [21, 30], [22, 32], [23, 29], [24, 17], [25, 3], [26, 10], [27, 11], [28, 4], [29, 23], [30, 21], [31, 6], [32, 22], [33, 9], [34, 12], [35, 18], [36, 36]]]

After

derived = [[1, 36], [[2, 19, 1], [3, 25, 1], [4, 28, 1], [5, 15, 1], [6, 31, 1], [7, 16, 1], [8, 14, 1], [9, 33, 1], [10, 26, 1], [11, 27, 1], [12, 34, 1], [13, 20, 1], [17, 24, 1], [18, 35, 1], [21, 30, 1], [22, 32, 1], [23, 29, 1]], 36, 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.

Step identifiers and dependencies

Rule: Wilson_GENERAL_inverse_pairing_and_p2_exception · Uses: G001, S109

Reverse checks

  • S110 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • 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 product36 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 product35 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 product34 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 product33 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 product32 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 product31 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 product30 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 product29 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 product28 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 product27 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 product26 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 product25 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 product24 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 product23 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 product22 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 product21 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 product20 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 product19 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 product18 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 product17 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 product16 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 product15 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 product14 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 product13 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 product12 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 product11 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 product10 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 product9 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 product8 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 product7 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 product6 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 product5 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 product4 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 product3 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 product2 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 product1 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 — 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.

Result

(37, [[1, 1], [2, 19], [3, 25], [4, 28], [5, 15], [6, 31], [7, 16], [8, 14], [9, 33], [10, 26], [11, 27], [12, 34], [13, 20], [14, 8], [15, 5], [16, 7], [17, 24], [18, 35], [19, 2], [20, 13], [21, 30], [22, 32], [23, 29], [24, 17], [25, 3], [26, 10], [27, 11], [28, 4], [29, 23], [30, 21], [31, 6], [32, 22], [33, 9], [34, 12], [35, 18], [36, 36]], [1, 36], [[2, 19, 1], [3, 25, 1], [4, 28, 1], [5, 15, 1], [6, 31, 1], [7, 16, 1], [8, 14, 1], [9, 33, 1], [10, 26, 1], [11, 27, 1], [12, 34, 1], [13, 20, 1], [17, 24, 1], [18, 35, 1], [21, 30, 1], [22, 32, 1], [23, 29, 1]], self_inverse_residues_1_and_p_minus_1, 36, Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 0a4050cbd599d9557abdddc655e1c2b4513c969bbc7b52ae8661c08330f77618
  • candidate record SHA-256: 896dccac465028cf525d6ebfa7e0082babaf94bb2f686a7aac28c8e2a1f09cc2
  • semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
  • question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON