Prime Square Perfect-Power Representation

Abstract

Every prime can be added to a square to obtain a perfect power with exponent at least two.

Theorem 1.1 (Every prime has a square-to-perfect-power representation).

$$\forall p\in \mathbb{N},\mathrm{Prime}\left(p\right)⟹\exists x,y,n\in \mathbb{N},2\le n\wedge x^2+p=y^n.$$

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

Resolves. Problems/oeis-a115039-prime-square-perfect-power-representation (proved) by D5/S3/Arith/PrimeSquarePerfectPowerRepresentation.result.

Citation. Cino Hilliard; Robert Israel (2006). OEIS A115039 and A115038, prime square perfect-power representation. URL: https://oeis.org/A115039.

Commentary.

For the prime two, five squared plus two is three cubed. Every odd prime has the form two times t plus one, and t squared plus that prime equals (t+1) squared. Both constructions use positive bases and an exponent at least two.

References

• Truth anchor: D5/S3/Arith/PrimeSquarePerfectPowerRepresentation.result