Values of binary partition function represented by a sum of three squares

TL;DR

This paper investigates the binary partition function with colored parts, characterizing when its values can be expressed as sums of three squares and determining the natural density of exceptions for specific parameters.

Contribution

It provides a novel characterization of the binary partition function's values in relation to sums of three squares, especially for cases where the number of colors is one less than a power of two.

Findings

For m=2^k - 1, the natural density of integers where b_m(n) is not a sum of three squares is 1/12 for k=1,2 and 1/6 for k≥3.

The paper characterizes exactly which n allow b_1(n) to be expressed as a sum of three squares, involving the Prouhet-Thue-Morse sequence.

Similar characterizations are obtained for b_{2^k-1}(n) in relation to sums of three squares.

Abstract

Let $m$ be a positive integer and $b_{m}(n)$ be the number of partitions of $n$ with parts being powers of 2, where each part can take $m$ colors. We show that if $m=2^{k}-1$, then there exists the natural density of integers $n$ such that $b_{m}(n)$ can not be represented as a sum of three squares and it is equal to $1/12$ for $k=1, 2$ and $1/6$ for $k\geq 3$. In particular, for $m=1$ the equation $b_{1}(n)=x^2+y^2+z^2$ has a solution in integers if and only if $n$ is not of the form $2^{2k+2}(8s+2t_{s}+3)+i$ for $i=0, 1$ and $k, s$ are non-negative integers, and where $t_{n}$ is the $n$th term in the Prouhet-Thue-Morse sequence. A similar characterization is obtained for the solutions in $n$ of the equation $b_{2^k-1}(n)=x^2+y^2+z^2$.