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Question 27707: Wilson Domain, Finite Testing, Fermat Comparison and the p=2 Exception Stay Separate

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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R0054-27707 · q-20261015-advance-20261118-r04-wilson-theorem-0082

Original question and givens

Use mode=fermat_confusion and exact integer n=17. Give (mode, n, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, Wilson_statement, Fermat_statement, theorem_identification_claim, boundary).

  • G001 — Use mode=fermat_confusion and exact integer n=17. Give (mode, n, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, Wilson_statement, Fermat_statement, theorem_identification_claim, boundary).

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-boundary]

After

derived = [[mode, n, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, Wilson_statement, Fermat_statement, theorem_identification_claim, boundary]]

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 · calculation

Before

Wilson boundary ORIGINAL prime ORIGINAL running factorial product1 := mul(1, 1)

After

Wilson boundary ORIGINAL prime 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, S001

S003 · implies

Before

source roles = [Wilson boundary ORIGINAL prime, 17, 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, S002

S004 · calculation

Before

Wilson boundary ORIGINAL prime ORIGINAL running factorial product2 := mul(1, 2)

After

Wilson boundary ORIGINAL prime 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, S003

S005 · implies

Before

source roles = [Wilson boundary ORIGINAL prime, 17, 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, S004

S006 · calculation

Before

Wilson boundary ORIGINAL prime ORIGINAL running factorial product3 := mul(2, 3)

After

Wilson boundary ORIGINAL prime 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, S005

S007 · implies

Before

source roles = [Wilson boundary ORIGINAL prime, 17, 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, S006

S008 · calculation

Before

Wilson boundary ORIGINAL prime ORIGINAL running factorial product4 := mul(6, 4)

After

Wilson boundary ORIGINAL prime 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, S007

S009 · implies

Before

source roles = [Wilson boundary ORIGINAL prime, 17, 4, 6, 24]

After

derived = [1, 7, Yes, Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S008

S010 · calculation

Before

Wilson boundary ORIGINAL prime ORIGINAL running factorial product5 := mul(7, 5)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product5 = 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, S009

S011 · implies

Before

source roles = [Wilson boundary ORIGINAL prime, 17, 5, 7, 35]

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, S010

S012 · calculation

Before

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

After

Wilson boundary ORIGINAL prime 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, S011

S013 · implies

Before

source roles = [Wilson boundary ORIGINAL prime, 17, 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, S012

S014 · calculation

Before

Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 := mul(6, 7)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 = 42

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 boundary ORIGINAL prime, 17, 7, 6, 42]

After

derived = [2, 8, Yes, Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S014

S016 · calculation

Before

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

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 = 64

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

source roles = [Wilson boundary ORIGINAL prime, 17, 8, 8, 64]

After

derived = [3, 13, Yes, Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S016

S018 · calculation

Before

Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 := mul(13, 9)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 = 117

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 boundary ORIGINAL prime, 17, 9, 13, 117]

After

derived = [6, 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, S018

S020 · calculation

Before

Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 := mul(15, 10)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 = 150

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 boundary ORIGINAL prime, 17, 10, 15, 150]

After

derived = [8, 14, Yes, Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S020

S022 · calculation

Before

Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 := mul(14, 11)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 = 154

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 boundary ORIGINAL prime, 17, 11, 14, 154]

After

derived = [9, 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, S022

S024 · calculation

Before

Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 := mul(1, 12)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 = 12

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · implies

Before

source roles = [Wilson boundary ORIGINAL prime, 17, 12, 1, 12]

After

derived = [0, 12, Yes, Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S024

S026 · calculation

Before

Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 := mul(12, 13)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 = 156

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

source roles = [Wilson boundary ORIGINAL prime, 17, 13, 12, 156]

After

derived = [9, 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, S026

S028 · calculation

Before

Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 := mul(3, 14)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 = 42

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 boundary ORIGINAL prime, 17, 14, 3, 42]

After

derived = [2, 8, Yes, Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S028

S030 · calculation

Before

Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 := mul(8, 15)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 = 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, S029

S031 · implies

Before

source roles = [Wilson boundary ORIGINAL prime, 17, 15, 8, 120]

After

derived = [7, 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, S030

S032 · calculation

Before

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

After

Wilson boundary ORIGINAL prime 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, S031

S033 · implies

Before

source roles = [Wilson boundary ORIGINAL prime, 17, 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, S032

S034 · implies

Before

source roles = [17, [[1, 1], [2, 2], [3, 6], [4, 7], [5, 1], [6, 6], [7, 8], [8, 13], [9, 15], [10, 14], [11, 1], [12, 12], [13, 3], [14, 8], [15, 1], [16, 16]], 16]

After

derived = [16, Yes, No]

Conditions: none

Wilson concerns(n-1)!;Fermat concerns a^(n-1) when n does not divide a.This branch supplies no a or power computation.The named comparison neither identifies the theorems nor adds an executed power test.

Step identifiers and dependencies

Rule: Wilson_and_Fermat_DISTINCT_named_statements_NO_unprovided_a_power · Uses: G001, S033

Reverse checks

  • S034 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S033 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S032 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S030 · premise_and_result_audit — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 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 boundary ORIGINAL prime ORIGINAL running factorial product14 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 boundary ORIGINAL prime ORIGINAL running factorial product13 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 boundary ORIGINAL prime ORIGINAL running factorial product12 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 boundary ORIGINAL prime ORIGINAL running factorial product11 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 boundary ORIGINAL prime ORIGINAL running factorial product10 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 boundary ORIGINAL prime ORIGINAL running factorial product9 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 boundary ORIGINAL prime ORIGINAL running factorial product8 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 boundary ORIGINAL prime ORIGINAL running factorial product7 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 boundary ORIGINAL prime ORIGINAL running factorial product6 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 boundary ORIGINAL prime ORIGINAL running factorial product5 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 boundary ORIGINAL prime ORIGINAL running factorial product4 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product3 with exact rational arithmetic, not rounded decimals.
  • 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product2 with exact rational arithmetic, not rounded decimals.
  • 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product1 with exact rational arithmetic, not rounded decimals.
  • 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

(fermat_confusion, 17, [[1, 1], [2, 2], [3, 6], [4, 7], [5, 1], [6, 6], [7, 8], [8, 13], [9, 15], [10, 14], [11, 1], [12, 12], [13, 3], [14, 8], [15, 1], [16, 16]], 16, 16, Yes, Wilson:(n-1)!==-1_mod_17, Fermat:a^(17-1)==1_mod_17_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed)

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: e4b418eaaa838c0f4a9309a948a15e8b164213771fb4d3f1669f7314296c6850
  • candidate record SHA-256: df02359ebf63e22bd97c89a1b3d8504a056d48e1a8415e822bf0e74d80f54c88
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  • Lean: NOT_RUN · human review: NOT_PERFORMED
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