A prime sigma-gcd is two or three

Abstract

A prime sigma-gcd is two or three.

Definition 1.1 (The exact OEIS conjecture).

$$\forall k\in \mathbb{N},\mathrm{Prime}\left(k-1\right)\Rightarrow \left(\mathrm{Prime}\left(k+1\right)\Rightarrow \left(\mathrm{Prime}\left(\gcd \left(k,\mathrm{sigma}\left(1,k\right)\right)\right)\Rightarrow \left(\gcd \left(k,\mathrm{sigma}\left(1,k\right)\right)=2\vee \gcd \left(k,\mathrm{sigma}\left(1,k\right)\right)=3\right)\right)\right)$$

Formalization. D5/S3/Arith/Robin/TwinPrimeSigmaGcdTwoOrThree.claim (✓ std3).

Source. Repository-derived.

Commentary.

The second A394757 conjecture asserts that a prime gcd of a twin-prime center and its divisor sum equals two or three. The claim retains both neighboring-prime hypotheses and quantifies over every natural center.

Theorem 1.2 (The universal implication).

$$\forall k\in \mathbb{N},\mathrm{Prime}\left(k-1\right)\Rightarrow \left(\mathrm{Prime}\left(k+1\right)\Rightarrow \left(\mathrm{Prime}\left(\gcd \left(k,\mathrm{sigma}\left(1,k\right)\right)\right)\Rightarrow \left(\gcd \left(k,\mathrm{sigma}\left(1,k\right)\right)=2\vee \gcd \left(k,\mathrm{sigma}\left(1,k\right)\right)=3\right)\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Arith/Robin/TwinPrimeSigmaGcdTwoOrThree.result (✓ std3). ∎

Resolves. Problems/oeis-a394757-sigma-gcd-two-or-three (proved) by D5/S3/Arith/Robin/TwinPrimeSigmaGcdTwoOrThree.result.

Source. Repository-derived.

Commentary.

If the divisor sum is even, two divides the prime gcd, which is therefore two. If it is odd, the shared square exclusion gives k=2t^2. Decompose nonzero t as 2^b r with r odd. Then k=2^(2b+1) r^2 and coprime multiplicativity expresses its divisor sum using the geometric sum for 2^(2b+1). The existing geometric-sum formula gives 2^(2b+2)-1, which is divisible by three because 4^(b+1) is congruent to one modulo three. The existing three_center theorem also gives three dividing k, so the prime gcd equals three. This settles preregistration #15004; infinitude remains open.

References

• Truth anchor: D5/S3/Arith/Robin/TwinPrimeSigmaGcdTwoOrThree.claim
• Truth anchor: D5/S3/Arith/Robin/TwinPrimeSigmaGcdTwoOrThree.result
• Dependency: D5/S3/Arith/Robin/TwinPrimeSigmaGcdThreeTwiceSquare