OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 27780: 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

Return to catalogue page 139

← Question derivation index · Original question batch

R0054-27780 · q-20261015-advance-20261118-r04-wilson-theorem-0030

Original question and givens

Use prime modulus p=11. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).

  • G001 — Use prime modulus p=11. 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 = [11, 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 = [11, 2]

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, S002

S004 · implies

Before

source roles = [11, 3]

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, S003

S005 · implies

Before

source roles = [11, 4]

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, S004

S006 · implies

Before

source roles = [11, 5]

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, S005

S007 · implies

Before

source roles = [11, 6]

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, S006

S008 · implies

Before

source roles = [11, 7]

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, S007

S009 · implies

Before

source roles = [11, 8]

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, S008

S010 · implies

Before

source roles = [11, 9]

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, S009

S011 · implies

Before

source roles = [11, 10]

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, S010

S012 · 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, S011

S013 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 11, 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, S012

S014 · 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, S013

S015 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 11, 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, S014

S016 · 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, S015

S017 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 11, 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, S016

S018 · 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, S017

S019 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 11, 4, 6, 24]

After

derived = [2, 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, S018

S020 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 := mul(2, 5)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 = 10

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, S019

S021 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 11, 5, 2, 10]

After

derived = [0, 10, 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, S020

S022 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(10, 6)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 60

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, S021

S023 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 11, 6, 10, 60]

After

derived = [5, 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, S022

S024 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(5, 7)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 35

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, S023

S025 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 11, 7, 5, 35]

After

derived = [3, 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, S024

S026 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(2, 8)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 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, S025

S027 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 11, 8, 2, 16]

After

derived = [1, 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, S026

S028 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(5, 9)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 45

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, S027

S029 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 11, 9, 5, 45]

After

derived = [4, 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, S028

S030 · calculation

Before

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(1, 10)

After

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 10

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, S029

S031 · implies

Before

source roles = [Wilson ORIGINAL prime factorial, 11, 10, 1, 10]

After

derived = [0, 10, 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, S030

S032 · implies

Before

source roles = [11, [[1, 1], [2, 6], [3, 4], [4, 3], [5, 9], [6, 2], [7, 8], [8, 7], [9, 5], [10, 10]]]

After

derived = [[1, 10], [[2, 6, 1], [3, 4, 1], [5, 9, 1], [7, 8, 1]], 10, 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, S031

Reverse checks

  • 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 — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S028 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S026 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S024 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S022 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 with exact rational arithmetic, not rounded decimals.
  • 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 — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 with exact rational arithmetic, not rounded decimals.
  • 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 product4 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 product3 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 product2 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 product1 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 — 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

(11, [[1, 1], [2, 6], [3, 4], [4, 3], [5, 9], [6, 2], [7, 8], [8, 7], [9, 5], [10, 10]], [1, 10], [[2, 6, 1], [3, 4, 1], [5, 9, 1], [7, 8, 1]], self_inverse_residues_1_and_p_minus_1, 10, 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: 89ad139f7e9fe25e1052123c77d4ea5fc5641afb22fa6f50466986257ff7a89f
  • candidate record SHA-256: 7c34ef79c364ac5c9d1e0fe93eb102083310674484f9abab1159bcbd2b57fbf0
  • semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
  • question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON