If you have some more information on m and n, there might be something more to say, but for general m and n, the only thing you can guarantee is 2.

Is this part of a larger problem?

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- March 10th 2010, 12:07 AM #1

- March 10th 2010, 08:11 AM #2

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- March 10th 2010, 12:38 PM #3

- March 10th 2010, 02:16 PM #4

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- March 10th 2010, 02:20 PM #5

- March 10th 2010, 03:35 PM #6

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Ok, I see now.

So your conjecture is that gcd(5^m + 7^m,5^n + 7^n) is sometimes 2 and sometimes 5^gcd(m,n) + 7^gcd(m,n) but never any other value, and you don't yet have a choice function to tell which one it is. That's an improvement on what I had to say. I said essentially that it's 2 or some multiple of 2.

For what it's worth, I can confirm your conjecture for all m,n less than 1000.

- March 21st 2010, 12:30 PM #7

- March 21st 2010, 10:38 PM #8

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I was waiting for the original poster to say something.

It depends on the factorization

which works if n/r is odd. So have a common factor if r is a common factor of n and m, and n/r and m/r are both odd.

So in the prime factorizations of n, m, and r, the power of 2 is the same. In fact, we should have:

if the prime factorizations of n and m contain the same power of 2.

I'm still foggy on the converse, though. Couldn't there be a common factor that's not of the form ?

- Hollywood