A result similar to Lagrange's theorem

TL;DR

This paper proves that every positive integer can be expressed as the sum of four generalized octagonal numbers with one odd, extending classical results on figurate numbers, and explores representations involving specific triples.

Contribution

It establishes a new universal representation theorem for positive integers using generalized octagonal numbers, including conditions on parity and specific coefficient triples.

Findings

Every positive integer is a sum of four generalized octagonal numbers with one odd.

Any nonnegative integer can be expressed as a combination involving specific triples (b,c,d).

The paper proposes conjectures for further research on generalized octagonal sums.

Abstract

Generalized octagonal numbers are those $p_8(x)=x(3x-2)$ with $x\in\mathbb Z$. In this paper we mainly show that every positive integer can be written as the sum of four generalized octagonal numbers one of which is odd. This result is similar to Lagrange's theorem on sums of four squares. Moreover, for $35$ triples $(b,c,d)$ with $1\le b\le c\le d$ (including $(2,3,4)$ and $(2,4,8)$), we prove that any nonnegative integer can be exprssed as $p_8(w)+bp_8(x)+cp_8(y)+dp_8(z)$ with $w,x,y,z\in\mathbb Z$. We also pose several conjectures for further research.