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The Sole Prime among the Integral Eighth-Quadratic Values

Abstract

The positive integral values of k(k+9)/8 have 17 as their only prime.

Definition 1.1 (Integral values with positive parameter).

Formalization. D5/S3/PrimeForms/OrlovskyEighthQuadraticSolePrime.IsTerm (✓ std3).

Citation. Vladimir Joseph Stephan Orlovsky; Bill McEachen (2009). OEIS A165719, Integers of the form k(k+9)/8*. URL: https://oeis.org/A165719.

Commentary.

A natural a is a term precisely when a positive natural k satisfies 8a=k(k+9). This equation selects the integral values without rounding.

Theorem 1.2 (The parameter classification and the sole prime).

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

Resolves. Problems/oeis-a165719-eighth-quadratic-integrality-and-sole-prime (proved) by D5/S3/PrimeForms/OrlovskyEighthQuadraticSolePrime.result.

Source. Repository-derived.

Acknowledgement. Vladimir Joseph Stephan Orlovsky; Bill McEachen (2009). OEIS A165719, Integers of the form k(k+9)/8*. URL: https://oeis.org/A165719.

Commentary.

For every positive k, integrality is equivalent to k+1=8m or k=8m for a positive natural m. The value 17 occurs at k=8 and is prime. Every other term is greater than one and is not prime. Reduction modulo eight leaves only residues seven and zero. In the first case a=k(m+1), with both factors greater than one. In the second case a=m(8m+9); m=1 gives 17, and every larger m gives a composite value.

References