关于数论函数$\sigma(n)$的一个注记

对于两个不相同的正整数$m$和$n$, 如果满足$\sigma(m)=\sigma(n)=m+n$, 则称之为一对亲和数, 这里$\sigma(n)=\sum_{d|n}d$.本文给出了$f(x,y)=x^{2^{x}}+y^{2^{x}}(x>y\geq{1},(x,y)=1)$不与任何正整数构成亲和数对的结论, 这里$x$,$y$具有不同的奇偶性, 即, 关于$z$的方程$\sigma(f(x,y))=\sigma(z)=f(x,y)+z$不存在正整数解.

A Note on Arithmetic Function $\sigma(n)$

Two distinct positive integers $m$ and $n$ are called amicable if $\sigma(m)=\sigma(n)=m+n$, where $\sigma(n)=\sum_{d\mid{n}}d$. This paper proves that $f(x,y)$ is not part of an amicable pair, where $f(x,y)=x^{2^{x}}+y^{2^{x}},x>y\geq{1},(x,y)=1$, one of $x$ and $y$ is odd number, the other is even. Hence, equation $\sigma(f(x,y))+\sigma(z)=f(x,y)+z$ has no positive integer solutions.