Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.06573

Integers in Sequences Generated by Recurrences with Non-Integer Coefficients

Max Lippmann
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we study the properties of linear second-order recurrences whose coefficients are non-integer rational numbers. We examine the set of indices at which the terms in the sequence are integers. Specifically, we prove two results. First, after fixing coefficients to satisfy certain coprimality assumptions, we show that there exist coprime integer initial conditions so that the set of integer terms of that sequence is as large as desired. Second, using the same coprimality assumptions, for each fixed pair of initial conditions, we give an explicit cutoff index after which no further terms are integral.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Max Lippmann. 2026-07-01. Integers in Sequences Generated by Recurrences with Non-Integer Coefficients. https://arxiv.org/abs/2607.06573

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. Liu proposed the conjecture \[ \sum_{\text{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \ge 2+\left(\frac{2r}{R}\right)^k,\qquad k>1, \] with the reverse inequality for $k<1$. We prove this conjecture by reducing it to an algebraic inequality for three positive variables with prescribed sum and product. We also determine the equality cases.

math.GM↗
arXiv2026

A quadratic critical-value conjecture for the fifth Bessel moment

We conjecture an explicit evaluation of the pure fifth Bessel moment $\int_0^\infty K_0(t)^5\,dt$ as a quadratic expression in the critical value $L(f,2)$ of the weight-three, level-60 newform $f$ (LMFDB orbit 60.3.b.a) identified in the twisted fifth-moment modularity theorem of Lim, Tu and Yu, with coefficients in $\mathbb{Q}(\sqrt{5})$ and the square taken before the real and imaginary parts. Directed interval computations, using no stored Bessel or $L$-values, bound the absolute discrepancy by $10^{-358}$. We prove three exact modular identities for $f$: the Petersson-norm formula $\langle f,f\rangle_{60} = \frac{3(5-\sqrt{5})}{2π^4}|L(f,2)|^2$, the coefficient-conjugation relation $L(f^σ,2) = κL(f,2)$ with explicit $κ\in \mathbb{Q}(\sqrt{5},i)$, and the twisted symmetric-square evaluation $L(χ_{-4}\mathrm{Sym}^2 f,2) = \sqrt{15}\,π^2 \langle f,f\rangle_{60}$, together with $L(χ_{-4}\mathrm{Sym}^2 f,3) = π^4\langle f,f\rangle_{60}/8$, in the full Euler-factor normalization of Lim, Tu and Yu. The last identity shows that the companion norm conjecture $D_{5,\mathrm{odd}} = \frac{3\sqrt{15}(5-\sqrt{5})}{2}|L(f,2)|^2$ is equivalent to the symmetric-square conjecture $D_{5,\mathrm{odd}} = π^2 L(χ_{-4}\mathrm{Sym}^2 f,2)$ of Lim, Tu and Yu, while the exact relation $D_{5,\mathrm{even}} = π^2 D_{5,\mathrm{odd}}/(2\sqrt{15})$ follows from Chuang's period formulas. Every Bessel-to-modular equality, including the individual-period formula, remains conjectural. Complete proofs, exact rational certificates and verification programs are included as ancillary files.

math.GM↗