derivation-mko-wilson-theorem-01 · ocme-new142-reader-adoption-20261021
For n>1, (n-1)!=-1 mod n exactly when n is prime; the full modulo-7 inverse table checks 6!=6 mod 7.
A general two-direction teaching argument plus full modulo-7 data and the p=2 branch; finite tests do not replace proof.
calcG1 — n is an integer greater than 1; use p in the prime direction, with p=7 as the example.d — proper positive divisorp — prime modulusSome rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.
calc · prerequisite — Assume exact integer arithmetic, division with remainder, and definitions of divisibility and gcd; every congruence has an explicit modulus. Explicitly assume coprime Bezout and Euclid's prime lemma p dividing ab implies p dividing a or b.mko_definition · cited — Qualify the integer domain n>1. For a prime modulus list the inverses of every nonzero residue and cancel the distinct inverse pairs. When p>2 the self-inverse residues are 1 and p−1; at p=2 they coincide. Exact factorial tables check the congruence at the specified modulus. Composite instances do not replace the converse proof, and a finite testing range does not prove the universal equivalence. Wilson factorial congruence and Fermat power congruence are different statements. · mko-wilson-theoremBefore
p\text{ prime},\quad1\le a<pAfter
\gcd(a,p)=1\Rightarrow\exists a^{-1}\pmod pConditions: none
Rule: calc · Uses: G1
Leaf basis: calc
A prime has only proper positive divisor 1, so every nonzero class is coprime and Bezout provides an inverse.
Before
a=a^{-1}\pmod pAfter
a^2\equiv1\Rightarrow p\mid(a-1)(a+1)\Rightarrow a\equiv1\text{ or }-1\pmod pConditions: none
Rule: calc · Uses: G1, S01
Leaf basis: calc
Euclid's lemma determines all self-inverse classes; for p>2,1 and -1 are distinct.
Before
p>2,\quad a\ne a^{-1}After
\{a,a^{-1}\}\text{ disjoint inverse pairs},\quad aa^{-1}\equiv1\pmod pConditions: none
Rule: calc · Uses: G1, S01, S02
Leaf basis: calc
Inversion is an involution, partitioning all non-self-inverse classes into disjoint two-element pairs.
Before
\prod_{a=1}^{p-1}a\text{ paired as above}After
(p-1)!\equiv1\cdot(-1)\cdot1\equiv-1\pmod p
Conditions: none
Rule: calc · Uses: G1, S02, S03
Leaf basis: calc
In the product of all nonzero classes, ordinary inverse pairs give 1 and self-inverses 1,-1 leave -1.
Before
p=2
After
(2-1)!=1\equiv-1\pmod2
Conditions: none
Rule: calc · Uses: G1, S01
Leaf basis: calc
At even prime 2,1 and -1 are the same single class; the formula holds directly.
Before
p=7,\quad a=1,2,3,4,5,6
After
a^{-1}=1,4,5,2,3,6\text{ respectively}Conditions: none
Rule: calc · Uses: G1, S01, S02
Leaf basis: calc
The complete six-entry inverse list gives pairs {1},{6},{2,4},{3,5}.
Before
j=1,2,3,4,5,6
After
j!\bmod7=1,2,6,3,1,6\text{ respectively};\quad6!=720\equiv6\pmod7Conditions: none
Rule: calc · Uses: G1, S04, S06
Leaf basis: calc
A complete incremental factorial table independently checks prime 7, with residue 6 representing -1.
Before
n>1\text{ composite},\quad1<d<n,\quad d\mid nAfter
d\mid(n-1)!
Conditions: none
Rule: calc · Uses: G1
Leaf basis: calc
A composite has a proper divisor d, which appears as a factor in 1 through n-1, so d divides the factorial.
Before
(n-1)!\equiv-1\pmod n,\quad d\mid n,\quad d\mid(n-1)!
After
d\mid((n-1)!+1)\text{ and }d\mid(n-1)!\Rightarrow d\mid1\text{ impossible}Conditions: none
Rule: calc · Uses: G1, S08
Leaf basis: calc
If a composite passed Wilson, subtraction would make d>1 divide 1, a contradiction, proving the converse.
Before
n>1,\quad\text{prime direction and composite contradiction}After
n\text{ prime}\iff(n-1)!\equiv-1\pmod nConditions: none
Rule: mko_definition · Uses: G1, S04, S05, S07, S09
Leaf basis: mko_definition
Both directions together with p=2 complete the equivalence; the modulo-7 table is only an instance certificate.
No substeps in this guide.
S10 · premise_audit · G1, S04, S05, S07, S09 — n=1 is outside the domain and cannot be added from a degenerate factorial residue.S09 · premise_audit · G1, S08 — This is a general contradiction argument, not a substitute based on several composite tests.S08 · premise_audit · G1 — No claim that every composite factorial is zero modulo n is needed; n=4 is covered too.S07 · premise_audit · G1, S04, S06 — Wilson uses a factorial rather than a Fermat base power.S06 · premise_audit · G1, S01, S02 — 2*4=8,3*5=15,6^2=36 all reduce to 1, checking every listed pair.S05 · premise_audit · G1, S01 — Do not count the coincident self-inverse representatives twice.S04 · premise_audit · G1, S02, S03 — This argument covers arbitrary odd primes rather than extrapolating the p=7 table.S03 · premise_audit · G1, S01, S02 — The inverse of a inverse is a, so no class is missed or paired twice.S02 · premise_audit · G1, S01 — Primality is essential for the zero-product implication; composite moduli need not satisfy it.S01 · premise_audit · G1 — The inverse is unique as a class: ab=ac=1 gives b=b(ac)=(ba)c=c.Every nonzero class must be paired or listed as self-inverse.
Separate prime pairing from the composite proper-divisor contradiction, without universal inference from tests.
reject — 3!=6 is 2 modulo 4, while -1 is 3.reject — 1=-1 modulo 2, giving one class.reject — Wilson concerns (p-1)!; Fermat concerns powers of a fixed base.For n>1, (n-1)!=-1 mod n exactly when n is prime; the full modulo-7 inverse table checks 6!=6 mod 7.
2 declared checks for this guide; only the listed expressions are covered.
b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f21bca817d48683970a4a4d8b31d62b9415855aa678ab3aa40682b18bfd6d1cdba2d4d999d4312d01eaa71a10b634018e935b72665829e46b09f492efd7ebc702SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEWSame-source full JSON (formulas, ASTs, controls and revision log)