论文标题
二元分区函数的值，由三个正方形表示
Values of binary partition function represented by a sum of three squares
论文作者
论文摘要
令$ m $为正整数，$ b_ {m}（n）$是$ n $的分区数，零件为2个，每个零件都可以采用$ m $颜色。我们表明，如果$ m = 2^{k} -1 $，则存在整数的自然密度$ n $，因此$ b_ {m}（n）$不能表示为三个正方形的总和，并且等于$ k = 1、2 $和$ 1/6 $的$ 1/12 $ for $ k = $ k \ geq geq for $ k \ geq 3 $ 3 $。特别是，对于$ m = 1 $，方程$ b_ {1}（n）= x^2+y^2+z^2 $在整数中具有解决方案，并且仅当$ n $不在$ 2^{2k+2}（8s+2t_+2t_ {s}} +3）+3）+i $ i = 0，$ i = 0，1 $和$ k，s $和s $ k，s $ k. $ t_ {n} $是Prouhet-thue-morse序列中的$ n $ th en $。在方程式$ b_ {2^k-1}（n）= x^2+y^2+z^2 $的$ n $中，获得了类似的表征。
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 $b_{2^k-1}(n)=x^2+y^2+z^2$.
