A165719 integrality and the sole prime

The OEIS entry states, verbatim:

%N A165719 Integers of the form k*(k+9)/8. %C A165719 Only one term is a prime number (17). Are all others composite? %F A165719 Conjecture: Valid k values are pairs of the form (8m-1,8m) for m=1,2,3.... If true, 17 is the only prime term. - Bill McEachen, May 06 2026 %A A165719 Vladimir Joseph Stephan Orlovsky, Sep 24 2009 %O A165719 1,1

For every positive natural k, 8 | k*(k+9) is equivalent to the existence of a positive natural m with k+1=8*m or k=8*m. A natural a is an integral term exactly when some positive k satisfies 8*a=k*(k+9). The value 17 is a term and a prime; every other term is greater than one and not prime. The example's positive-parameter restriction is retained. The equations avoid natural-number floor division and truncated subtraction.

Not claimed: a result for nonpositive parameters, a new method of prime classification, any assertion about arbitrary quadratic forms, an exhaustive literature search, or priority beyond the settlement of these named assertions in the searched scope.

Orlovsky submitted the sequence in September 2009. The sole-prime question is present in archived revision #1 (June 1 2010), and McEachen's parameter conjecture was added on May 6 2026. The proof is elementary; the novelty is the settlement of the named assertions, not the technique.

Revision #24 (May 14 2026 00:56:29) still carries both assertions without a settlement comment or proof link. There is no %D bibliography; its only %H is an internal linear-recurrence index. Dale's first-million-term check is finite evidence, not a universal proof. Repository and pinned Mathlib statement-shape searches, online Loogle queries for the divisibility and prime-product shapes, and GitHub Lean-code searches returned no exact settlement. The preregistration's bounded external web search also found no settlement. This is not-found-in-searched-scope, not an exhaustive claim.

Reduction modulo eight shows that only residues zero and seven make the product divisible by eight. If k+1=8*m, cancellation gives a=k*(m+1); both factors exceed one. If k=8*m, cancellation gives a=m*(8*m+9). The case m=1 gives 17; every other positive m makes both factors exceed one. The universal conclusion follows from product bounds and nonprimality of a product whose factors differ from one.

A positive parameter with an integral value but residue outside zero and seven modulo eight, or a prime integral term different from 17, would contradict the respective conjunct of result.

The Lean module is D5/S3/PrimeForms/OrlovskyEighthQuadraticSolePrime.lean. Its single theorem contains the full parameter classification, the prime term 17, and the composite characterization of every other term. The proof uses Mathlib's modular arithmetic, product bounds, cancellation, and Nat.not_prime_mul; it introduces no additional mathematical assumption.

theorem: both named assertions are proved for all positive natural parameters and all integral terms, not just a bounded initial segment. The resolution kind is proved, not refuted.

Literature completeness outside the named searches is unverified. Google Scholar and MathSciNet were not searched; no exhaustive literature or priority claim is made. The preregistration's web and revision-history readings are attributed to the orchestrator; the current OEIS text and Lean-library searches were independently checked by the implementation seat.