Sums of seven octahedral numbers

TL;DR

This paper proves that all sufficiently large numbers can be expressed as sums of seven positive values of certain cubic polynomials, including octahedral numbers, reducing an open problem to finite computation.

Contribution

It establishes a general result for sums of seven positive values of cubic polynomials, specifically solving the open problem for octahedral numbers.

Findings

Every number greater than e^{10^7} is a sum of seven positive octahedral numbers.

Reduces the open problem of representing large numbers as sums of octahedral numbers to finite computation.

Provides a general framework for representing large numbers as sums of polynomial values.

Abstract

We show that for a large class of cubic polynomials $f$, every sufficiently large number can be written as a sum of seven positive values of $f$. As a special case, we show that every number greater than $e^{10^7}$ is a sum of seven positive octahedral numbers, where an octahedral number is a number of the form $\frac{2x^3+x}{3}$, reducing an open problem due to Pollock to a finite computation.