Formulas for the Generalized Frobenius Number of Triangular Numbers

TL;DR

We study the generalized Frobenius number for three consecutive triangular numbers, $\mathtt{g}(t_n,t_{n+1},t_{n+2};s)$, and derive explicit closed forms valid for all $s\ge0$. The method combines a Beck–Kifer reduction with a parity-sensitive decomposition using auxiliary data $ (x_s^{\mathrm{even}},y_s^{\mathrm{even}})$ and $(x_s^{\mathrm{odd}},y_s^{\mathrm{odd}})$, plus index bounds $N_s^{\mathrm{even}}$ and $N_s^{\mathrm{odd}}$. This yields, for $s= k(k+1)+i$, exact formulas depending on the parity of $n$ and whether $s+1$ lies in the exceptional set $\mathbb{B}=\{k^2, k(k+1)\}$, and provides a complete proof of Komatsu's conjecture. The results give precise, parity-dependent expressions for the generalized Frobenius number in the triangular-number setting and advance understanding of restricted representations in this Diophantine context.

Abstract

For $ k \geq 2 $, let $ A = (a_{1}, a_{2}, \ldots, a_{k}) $ be a $k$-tuple of positive integers with $\gcd(a_{1}, a_2, \ldots, a_k) = 1$. For a non-negative integer $s$, the generalized Frobenius number of $A$, denoted as $\mathtt{g}(A;s) = \mathtt{g}(a_1, a_2, \ldots, a_k;s)$, represents the largest integer that has at most $s$ representations in terms of $a_1, a_2, \ldots, a_k$ with non-negative integer coefficients. In this article, we provide a formula for the generalized Frobenius number of three consecutive triangular numbers, $\mathtt{g}(t_{n}, t_{n+1}, t_{n+2};s) $, valid for all $s \geq 0$ where $t_n$ is given by $\binom{n+1}{2}$. Furthermore, we present the proof of Komatsu's conjecture

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Corollary 3
• Definition 4
• Theorem 5
• Lemma 6
• Lemma 7
• proof
• Lemma 8
• Lemma 9