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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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Thomas Andrews
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1 Answers1

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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.

Thomas Andrews
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Vladimir
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  • Yeah, that was exactly what I was looking for, thanks! – Thomas Andrews Jun 04 '14 at 19:42
  • You are welcome. – Vladimir Jun 04 '14 at 19:43
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    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}\ $ – Bill Dubuque Jun 04 '14 at 22:57