derivation-mko-fermat-little-theorem-01 · ocme-new142-reader-adoption-20261021
(-2)^6 is congruent to 1 modulo 7; for multiples of 7, only the congruence a^7≡a modulo 7 applies; passing a Fermat test does not prove primality.
Explicit prime modulus 7, full multiplication permutation and exact squaring, with composite 341 separately.
calcG1 — p=7,a=-2, normalized to 5 modulo 7; the theorem statements are congruences, not integer equalities.p — qualified prime modulusU_7 — all nonzero residues modulo 7calc · prerequisite — Assume exact integer arithmetic, division with remainder, and definitions of divisibility and gcd; every congruence has an explicit modulus. Explicitly assume the complete trial-division-to-square-root primality criterion.mko_definition · cited — The theorem modulus is an explicit prime p≥2. The coprime form requires p not dividing a; the all-integer form also handles p dividing a without canceling a. Exact modular powers normalize negative bases. Composite examples such as561 show that passing a Fermat test does not prove primality. All residue equalities here have the declared modulo-p scope. · mko-fermat-little-theoremBefore
p=7,\quad a=-2
After
7\text{ prime},\quad a\equiv5\pmod7,\quad7\nmid aConditions: none
Rule: calc · Uses: G1
Leaf basis: calc
A composite 7 would have a prime divisor at most sqrt(7); only 2 is possible and does not divide 7. The base is a unit.
Before
U_7=\{1,2,3,4,5,6\}After
5U_7\pmod7=(5,3,1,6,4,2)
Conditions: none
Rule: calc · Uses: G1, S01
Leaf basis: calc
Multiplying every nonzero residue by 5 produces each exactly once, a complete permutation.
Before
\prod_{u\in U_7}(5u)\equiv\prod_{u\in U_7}u\pmod7After
5^6\cdot720\equiv720\pmod7,\quad720\equiv6\pmod7
Conditions: none
Rule: calc · Uses: G1, S02
Leaf basis: calc
Permutation preserves the product; factoring out six copies of 5 yields the product to cancel.
Before
5^6\cdot6\equiv6\pmod7
After
5^6\equiv1\pmod7\Rightarrow(-2)^6\equiv1\pmod7
Conditions: none
Rule: calc · Uses: G1, S01, S03
Leaf basis: calc
Multiply by 6 to cancel a unit and restore the original negative representative.
Before
5^2\pmod7
After
5^2\equiv4,\quad5^4\equiv2,\quad5^6\equiv2\cdot4\equiv1\pmod7
Conditions: none
Rule: calc · Uses: G1, S04
Leaf basis: calc
Independent exact squaring and multiplication recheck the same result without floating-point powers.
Before
a=7b,\quad b\in\mathbb Z
After
a^7\equiv0\equiv a\pmod7,\quad a^6\equiv0\not\equiv1\pmod7
Conditions: none
Rule: calc · Uses: G1, S01
Leaf basis: calc
If 7 divides a, the all-integer form still holds; the coprime form does not apply and a cannot be canceled.
Before
n=341=11\cdot31,\quad2^{10}=1024After
1024=3\cdot341+1\Rightarrow2^{340}=(2^{10})^{34}\equiv1\pmod{341}Conditions: none
Rule: calc · Uses: G1
Leaf basis: calc
Composite 341 passes the base-2 Fermat congruence, directly refuting a primality converse.
Before
p=7:\quad(-2)^6\equiv1,\quad a=7b:\ a^7\equiv a
After
\text{coprime and all-integer forms have different domains}Conditions: none
Rule: mko_definition · Uses: G1, S04, S05, S06, S07
Leaf basis: mko_definition
Keep the two forms and composite counterexample separate, retaining the modulus on every equality.
No substeps in this guide.
S08 · premise_audit · G1, S04, S05, S06, S07 — a^p=a is not an identity in Z; for example 2^7=128 is not 2.S07 · premise_audit · G1 — 11 and 31 exceed 1, certifying compositeness; this does not refute the prime direction.S06 · premise_audit · G1, S01 — 0^6 has a positive exponent and is zero, not a zero-division or 0^0 issue.S05 · premise_audit · G1, S04 — Every product is reduced modulo 7; huge integer expansion is unnecessary.S04 · premise_audit · G1, S01, S03 — Cancellation relies on an invertible product and cannot be copied to zero.S03 · premise_audit · G1, S02 — The remainder 6 of 720 is nonzero and 6*6=1 modulo 7, hence invertible.S02 · premise_audit · G1, S01 — All six inputs and outputs are listed, not a sample.S01 · premise_audit · G1 — Primality and a nonzero base are qualified first, not inferred from passing Fermat.If p divides a, a cannot be canceled to obtain the coprime form.
Certify primality before applying the theorem; a passing congruence does not certify primality.
reject — The base is zero modulo 7 and positive powers stay zero.reject — 341=11*31 is composite.reject — For a=2,128 differs from 2; only their difference 126 is divisible by 7.(-2)^6 is congruent to 1 modulo 7; for multiples of 7, only the congruence a^7≡a modulo 7 applies; passing a Fermat test does not prove primality.
3 declared checks for this guide; only the listed expressions are covered.
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