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Question 27786: 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

R0054-27786 · q-20261015-advance-20261118-r04-wilson-theorem-0036

Original question and givens

Use prime modulus p=31. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).

  • G001 — Use prime modulus p=31. 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 = [31, 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 = [31, 2]

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, S002

S004 · implies

Before

source roles = [31, 3]

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, S003

S005 · implies

Before

source roles = [31, 4]

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, S004

S006 · implies

Before

source roles = [31, 5]

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, S005

S007 · implies

Before

source roles = [31, 6]

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, S006

S008 · implies

Before

source roles = [31, 7]

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, S007

S009 · implies

Before

source roles = [31, 8]

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, S008

S010 · implies

Before

source roles = [31, 9]

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, S009

S011 · implies

Before

source roles = [31, 10]

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, S010

S012 · implies

Before

source roles = [31, 11]

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, S011

S013 · implies

Before

source roles = [31, 12]

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, S012

S014 · implies

Before

source roles = [31, 13]

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, S013

S015 · implies

Before

source roles = [31, 14]

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, S014

S016 · implies

Before

source roles = [31, 15]

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, S015

S017 · implies

Before

source roles = [31, 16]

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, S016

S018 · implies

Before

source roles = [31, 17]

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, S017

S019 · implies

Before

source roles = [31, 18]

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, S018

S020 · implies

Before

source roles = [31, 19]

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, S019

S021 · implies

Before

source roles = [31, 20]

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, S020

S022 · implies

Before

source roles = [31, 21]

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, S021

S023 · implies

Before

source roles = [31, 22]

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, S022

S024 · implies

Before

source roles = [31, 23]

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, S023

S025 · implies

Before

source roles = [31, 24]

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, S024

S026 · implies

Before

source roles = [31, 25]

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, S025

S027 · implies

Before

source roles = [31, 26]

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, S026

S028 · implies

Before

source roles = [31, 27]

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, S027

S029 · implies

Before

source roles = [31, 28]

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, S028

S030 · implies

Before

source roles = [31, 29]

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, S029

S031 · implies

Before

source roles = [31, 30]

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, S030

S032 · 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, S031

S033 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 31, 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, S032

S034 · 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, S033

S035 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 31, 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, S034

S036 · 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, S035

S037 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 31, 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, S036

S038 · 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, S037

S039 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 31, 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, S038

S040 · 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, S039

S041 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 31, 5, 24, 120]

After

derived = [3, 27, Yes, Yes]

Conditions: none

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

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 product6 := mul(27, 6)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 162

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, 31, 6, 27, 162]

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, S042

S044 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 31, 7, 7, 49]

After

derived = [1, 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, S044

S046 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(18, 8)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 144

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, 31, 8, 18, 144]

After

derived = [4, 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, S046

S048 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(20, 9)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 180

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 31, 9, 20, 180]

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, S048

S050 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(25, 10)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 250

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, 31, 10, 25, 250]

After

derived = [8, 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, S050

S052 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(2, 11)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 31, 11, 2, 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.

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 product12 := mul(22, 12)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 264

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, 31, 12, 22, 264]

After

derived = [8, 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, S054

S056 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(16, 13)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 208

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, 31, 13, 16, 208]

After

derived = [6, 22, Yes, Yes]

Conditions: none

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

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 product14 := mul(22, 14)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 308

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, 31, 14, 22, 308]

After

derived = [9, 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, S058

S060 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(29, 15)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 435

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, 31, 15, 29, 435]

After

derived = [14, 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, S060

S062 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(1, 16)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 16

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 31, 16, 1, 16]

After

derived = [0, 16, Yes, Yes]

Conditions: none

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

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 product17 := mul(16, 17)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 272

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S063

S065 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 31, 17, 16, 272]

After

derived = [8, 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, S064

S066 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(24, 18)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 432

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, 31, 18, 24, 432]

After

derived = [13, 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, S066

S068 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(29, 19)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 551

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, 31, 19, 29, 551]

After

derived = [17, 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, S068

S070 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(24, 20)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 480

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, 31, 20, 24, 480]

After

derived = [15, 15, Yes, Yes]

Conditions: none

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

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 product21 := mul(15, 21)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 315

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, 31, 21, 15, 315]

After

derived = [10, 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.

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 product22 := mul(5, 22)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 110

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, 31, 22, 5, 110]

After

derived = [3, 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, S074

S076 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(17, 23)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 391

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S075

S077 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 31, 23, 17, 391]

After

derived = [12, 19, Yes, Yes]

Conditions: none

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

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 product24 := mul(19, 24)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 456

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, 31, 24, 19, 456]

After

derived = [14, 22, Yes, Yes]

Conditions: none

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

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 product25 := mul(22, 25)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 550

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S079

S081 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 31, 25, 22, 550]

After

derived = [17, 23, Yes, Yes]

Conditions: none

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

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 product26 := mul(23, 26)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 598

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, 31, 26, 23, 598]

After

derived = [19, 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, S082

S084 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(9, 27)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 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, S083

S085 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 31, 27, 9, 243]

After

derived = [7, 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, S084

S086 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(26, 28)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 728

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, 31, 28, 26, 728]

After

derived = [23, 15, Yes, Yes]

Conditions: none

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

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 product29 := mul(15, 29)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 435

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, 31, 29, 15, 435]

After

derived = [14, 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, S088

S090 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(1, 30)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S089

S091 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 31, 30, 1, 30]

After

derived = [0, 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 · implies

Before

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

After

derived = [[1, 30], [[2, 16, 1], [3, 21, 1], [4, 8, 1], [5, 25, 1], [6, 26, 1], [7, 9, 1], [10, 28, 1], [11, 17, 1], [12, 13, 1], [14, 20, 1], [15, 29, 1], [18, 19, 1], [22, 24, 1], [23, 27, 1]], 30, 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, S091

Reverse checks

  • S092 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • 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 product30 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 product29 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 product28 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 product27 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 product26 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 product25 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 product24 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 product23 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 product22 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 product21 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 product20 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 product19 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 product18 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 product17 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 product16 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 product15 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 product14 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 product13 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 product12 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 product11 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 product10 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 product9 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 product8 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 product7 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 product6 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 product5 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 product4 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 product3 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 product2 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 product1 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 — 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

(31, [[1, 1], [2, 16], [3, 21], [4, 8], [5, 25], [6, 26], [7, 9], [8, 4], [9, 7], [10, 28], [11, 17], [12, 13], [13, 12], [14, 20], [15, 29], [16, 2], [17, 11], [18, 19], [19, 18], [20, 14], [21, 3], [22, 24], [23, 27], [24, 22], [25, 5], [26, 6], [27, 23], [28, 10], [29, 15], [30, 30]], [1, 30], [[2, 16, 1], [3, 21, 1], [4, 8, 1], [5, 25, 1], [6, 26, 1], [7, 9, 1], [10, 28, 1], [11, 17, 1], [12, 13, 1], [14, 20, 1], [15, 29, 1], [18, 19, 1], [22, 24, 1], [23, 27, 1]], self_inverse_residues_1_and_p_minus_1, 30, 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: 2e1b314590ae01c803bc9f2d2291bf3c29d0bba5b0be9fa5f23b8234f590d76d
  • candidate record SHA-256: 7452bf82951900665b24953bb56f9da9401eda1e936f8b8d793c49c8d08ed8cd
  • semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
  • question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON