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If $f:\mathbb N\to\mathbb Z$ satisfies:

$$\forall n,m\in\mathbb N\,, n+m\mid f(n)+f(m)$$

How to show that this implies:

$$\forall n,m\in\mathbb N,\,n-m\mid f(n)-f(m)?$$

I was almost incidentally able to prove this by classifying such functions, but that seems circuitous for such a result. Is there a proof that is (more) direct?

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This is actually pretty easy. Let $n>m$, and take an $N$ such that $N(n-m)>m$. Set $a=N(n-m)-m$. Then $$ m+a=N(n-m),\qquad n+a=m+a+n-m=(N+1)(n-m). $$ Now $$ f(n)-f(m)=f(n)+f(a)-(f(m)+f(a)), $$ but by assumption $n-m\mid f(m)+f(a)$ and $n-m\mid f(n)+f(a)$, and we are done.

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  • $\begingroup$ Yeah, that was exactly what I was looking for, thanks! $\endgroup$ – Thomas Andrews Jun 4 '14 at 19:42
  • $\begingroup$ You are welcome. $\endgroup$ – Vladimir Jun 4 '14 at 19:43
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    $\begingroup$ i.e. $\ \ \ \begin{eqnarray} && n\!-\!m\!\!\!&&\mid \ n\!+\!a\!\!\!&&\mid \ \color{#0a0}{f_{\large n}\!+f_{\large a}}\\ \Rightarrow &&n\!-\!m\!\!\!&&\mid m\!+\!a\!\!\!&&\mid \color{#c00}{f_{\large m}\!+f_{\large a}}\end{eqnarray}\Bigg\}\,\Rightarrow\ n\!-\!m\mid (\color{#0a0}{f_{\large n}\!+f_{\large a}})-(\color{#c00}{f_{\large m}\!+f_{\large a}}) = f_{\large n}\!-f_{\large m}\ $ $\endgroup$ – Bill Dubuque Jun 4 '14 at 22:57

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