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Question 27779: 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

R0054-27779 · q-20261015-advance-20261118-r04-wilson-theorem-0029

Original question and givens

Use prime modulus p=7. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).

  • G001 — Use prime modulus p=7. 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 = [7, 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 = [7, 2]

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, S002

S004 · implies

Before

source roles = [7, 3]

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, S003

S005 · implies

Before

source roles = [7, 4]

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, S004

S006 · implies

Before

source roles = [7, 5]

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, S005

S007 · implies

Before

source roles = [7, 6]

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, S006

S008 · 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, S007

S009 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 7, 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, S008

S010 · 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, S009

S011 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 7, 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, S010

S012 · 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, S011

S013 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 7, 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, S012

S014 · 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, S013

S015 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 7, 4, 6, 24]

After

derived = [3, 3, Yes, Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S014

S016 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 := mul(3, 5)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 = 15

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, S015

S017 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 7, 5, 3, 15]

After

derived = [2, 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, S016

S018 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(1, 6)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 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, S017

S019 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 7, 6, 1, 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, S018

S020 · implies

Before

source roles = [7, [[1, 1], [2, 4], [3, 5], [4, 2], [5, 3], [6, 6]]]

After

derived = [[1, 6], [[2, 4, 1], [3, 5, 1]], 6, 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, S019

Reverse checks

  • 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 — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S016 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S014 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S012 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S010 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S008 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S006 · premise_and_result_audit — 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

(7, [[1, 1], [2, 4], [3, 5], [4, 2], [5, 3], [6, 6]], [1, 6], [[2, 4, 1], [3, 5, 1]], self_inverse_residues_1_and_p_minus_1, 6, 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: 0ef78b541b9caee28ca0f3c89acbb53e3abf98f7a50bbe00e62cb8c9ef266eea
  • candidate record SHA-256: 4de4020dca268f7958f906b6bebeb850fba174209cd63a42375cabbfd1893240
  • semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
  • question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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