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I believe I've found all Amicable Pairs $(n, m)$ such that n is in the closed interval $[1, 34000000]$. Here's my list.

By inspection I believe that for all $n$ in the list, $n$ is odd iff its last digit is $5$. Can it be proved that this is true for all Amicable Pairs?

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2 Answers

up vote 7 down vote accepted

I think the first counterexample $(n,m)$ with $n < m$ where $n$ is odd and not congruent to $5$ mod $10$ is $(34765731, 36939357)$, which by misfortune is just outside your search region.

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Damn. You're right. Thanks. BTW I'm rusty. Could you spell out the terminology in "not congruent to 5 mod 10"? Thanks. –  NotSuper Nov 15 '10 at 15:02
In this case, just "doesn't end w/5", but seems less base-related.. :^) See en.wikipedia.org/wiki/Modular_arithmetic. –  DSM Nov 15 '10 at 15:05
Thanks. Good article. It's coming back, slowly. –  NotSuper Nov 15 '10 at 15:17
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How about 87633 and 69615? Found 87633 in OEIS A063990

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I neglected to say that n < m. The n of the pair (69615, 87633) is odd and ends in '5'. –  NotSuper Nov 15 '10 at 14:57
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