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An unavoidable set of divisors for the nearest integers to ξ(3/2)^n

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Ralf Stephan ↗

Version 1 / Oct 14, 2026 / CC0 1.0

Abstract

We prove that for every real ξ ̸= 0 the sequence of nearest integers [ξ(3/2)^n + 1/2], n ∈ N, contains infinitely many elements divisible by 2 or by 5. In particular, for ξ > 0 infinitely many of these nearest integers are composite. This is the nearest-integer counterpart of the theorem of Forman and Shapiro on [ξ(3/2)^n], for which Dubickas and Novikas give the unavoidable set {2, 5, 7, 11}; for nearest integers two primes suffice. The proof follows the method of Dubickas and Novikas: if 2 and 5 divide only finitely many terms, the sequence of operations is forced to become (A^2B^2)^∞, which is impossible by their Lemma 2. All results are machine-verified in Lean 4.

Provenance statement

Author collected the literature. Author chose the problem and directed LLM to find a solution and formalize it. Author confirms the Lean-checked theorem corresponds to the main theorem.

Tools used

Anthropic
Claude OpusVersion 5.5

References

  1. A. Dubickas and A. Novikas, Integer parts of powers of rational numbers , Math.\ Z.\ 251 (2005), 635--648.
  2. W. Forman and H. N. Shapiro, An arithmetic property of certain rational powers , Comm.\ Pure Appl.\ Math.\ 20 (1967), 561--573.
  3. A. Novikas, Composite numbers in the sequences of integers , doctoral dissertation, Vilnius University, 2012.

Version history

  1. v1Submitted by Ralf StephanInitial depositCurrentOct 14, 2026