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GCD-square reformulation of coprime fab rescue

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Let p≡1 mod4 be prime and suppose coprime positive a,b give a fab certificate with divisor k:

Source in the repository

Status: proved exact construction and converse on the coprime fab range

Date: 2026-08-15

Depends on: FAB-COPRIME-DIVISOR-CRITERION.md, FAB-HARD-NONRESIDUE-BRIDGE.md

Claim boundary: this is a new exact reformulation/construction. It does not prove that the required pair always exists, and therefore does not prove Erdős–Straus.

1. From a coprime fab certificate to two 3 mod 4 integers

Let p≡1 mod4 be prime and suppose coprime positive a,b give a fab certificate with divisor k:

k\mid a+bp, \qquad k\equiv-p\pmod{4ab}.

Write

\boxed{p+k=4abc}

and

q=\frac{a+bp}{k}.

The auxiliary factorization from FAB-HARD-NONRESIDUE-BRIDGE.md gives

k(q+b)=a(1+4b^2c).

Since gcd(a,k)=1, put

\boxed{kd=1+4b^2c.}

Because k≡3 mod4 and the right side is 1 mod4,

\boxed{d\equiv3\pmod4.}

Define

\boxed{M=kd-1=4b^2c.}

Then

p+k=4abc.

Since gcd(a,b)=1,

\gcd(p+k,M) = \gcd(4abc,4b^2c) = \boxed{4bc}.

Call this gcd G. Then

\boxed{G=4bc,} \qquad \boxed{M=4b^2c.}

Consequently

\boxed{G^2=4cM,}

so in particular

\boxed{4M\mid G^2.}

Moreover the original parameters are recovered exactly:

\boxed{ a=\frac{p+k}{G}, \qquad b=\frac{M}{G}, \qquad c=\frac{G^2}{4M}.}

Thus every coprime fab certificate produces two 3 mod 4 integers k,d for which the two numbers

p+k, \qquad kd-1

have unusually large square overlap.

2. Converse construction from a gcd-square overlap

Now start with an odd positive integer p≡1 mod4 and positive integers

\boxed{k\equiv d\equiv3\pmod4.}

Put

M=kd-1, \qquad G=\gcd(p+k,M).

Assume the exact divisibility

\boxed{4M\mid G^2.}

Define

\boxed{ a=\frac{p+k}{G}, \qquad b=\frac{M}{G}, \qquad c=\frac{G^2}{4M}.}

Then a,b,c are positive integers, and because G is the full gcd,

\boxed{\gcd(a,b)=1.}

The definitions give

4abc =4\frac{p+k}{G}\frac{M}{G}\frac{G^2}{4M} =p+k,

hence

\boxed{p+k=4abc.}

They also give

4b^2c =4\frac{M^2}{G^2}\frac{G^2}{4M} =M=kd-1,

so

\boxed{kd=1+4b^2c.}

Set

\boxed{q=ad-b.}

It is positive, because

q =\frac{d(p+k)-(kd-1)}{G} =\boxed{\frac{dp+1}{G}}>0.

Finally,

a+bp =a+b(4abc-k) =a+4ab^2c-bk =a+a(kd-1)-bk =k(ad-b) =kq.

Therefore

\boxed{k\mid a+bp.}

And since p+k=4abc,

\boxed{k\equiv-p\pmod{4ab}}.

Thus the coprime divisor criterion is satisfied whenever its standard size hypotheses are desired. More strongly, the identities above directly produce the positive decomposition

\boxed{ \frac4p = \frac1{abc} + \frac1{aqc} + \frac1{bpqc}, }

so the construction itself does not need a separate bounded-parameter argument.

3. GCD-square rescue theorem

Theorem

Let p≡1 mod4 be a positive odd integer. If there exist positive

k\equiv d\equiv3\pmod4

such that, with

M=kd-1, \qquad G=\gcd(p+k,M),

one has

\boxed{4M\mid G^2,}

then p satisfies the Erdős–Straus equation.

Every coprime fab certificate produces such a pair (k,d), with the stronger identity

\boxed{G^2=4cM}

for its leftover parameter c.

4. Structural interpretation

The all-prime divisor problem can therefore be attacked as a square-overlap problem:

\boxed{ \text{find }k,d\equiv3\pmod4 \text{ such that } kd-1 \text{ is supported deeply enough inside }p+k. }

The condition is not merely that the two numbers share a factor. It requires their full gcd to contain at least half of every prime-power valuation of 4(kd-1):

2v_\ell(G)\ge v_\ell(4M) \qquad\text{for every prime }\ell.

This valuation form may be better suited to a descent than the original divisor-in-one-residue-class statement.

Two boundary cases recover familiar shifted-factor mechanisms:

  • if M | p+k, then G=M, and the condition reduces to 4|M; since k,d≡3 mod4, this is automatic. Thus any factorization
kd-1\mid p+k

is an immediate rescue;

  • taking M=p+k gives
k(d-1)=p+1,

which contains the classical p+1 / 3 mod4 divisor spine as a special case.

The new target is to determine whether the exact hard-prime factor restrictions force such a square overlap for some controlled (k,d), rather than continuing to enumerate unrelated (a,b) pairs.