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A Level number theory help

If m and n are positive integers and m is odd, prove that (2^m)-1 and (2^n)+1 are coprime.

(Hint: consider (2^m)^n and (2^n)^m).

Any help is appreciated.
I can't quite understand your notation, is this 2 to the power of m and then to the power of negative 1, and then 2 to the power of n+1?
Original post by 0-)
If m and n are positive integers and m is odd, prove that (2^m)-1 and (2^n)+1 are coprime.

(Hint: consider (2^m)^n and (2^n)^m).

Any help is appreciated.


What have you tried so far?


@Ostiomanic It is, 2 to the power m, minus one. And then 2 to the power of n, plus 1. Otherwise it doesn't make sense as a question.
(edited 4 years ago)
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Original post by zetamcfc
What have you tried so far?


@Ostiomanic It is, 2 to the power m, minus one. And then 2 to the power of n, plus 1. Otherwise it doesn't make sense as a question.

I looked at (2^m)^n and (2^n)^m and tried to use methods in the course like Fermat's Little Theorem but I couldn't make any progress. Can I have a hint to start me off?
Original post by 0-)
I looked at (2^m)^n and (2^n)^m and tried to use methods in the course like Fermat's Little Theorem but I couldn't make any progress. Can I have a hint to start me off?

Suppose gcd of the two is >1. Then what do we know about a prime dividing both? Look at the congruences 2^m - 1 mod p and 2^n +1 mod p. What can you say about the order of 2 mod p?
(edited 4 years ago)

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