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(GCD/prime problem) positive integers m and n, prove that (2^m -1, 2^n +1) = 1

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given two positive integers, m and n.

m is an odd number.

prove that (2^m -1, 2^n +1) = 1

what i have in my notes is that: (a, b) = 1  --> gcd is 1 --> that means that a and b are coprime.

So I have to prove that 2^m -1 and 2^n +1  are both prime numbers, aka prove that their gcd is 1.

I'm just not sure how to do that, any help is appreciated!

Feb 9, 2019

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$$\text{what you think you have to prove can't be right because}\\ 2^6+1 = 65 = 5 \cdot 13 \text{ isn't prime, and}\\ 2^9-1 = 511 = 7\cdot 73 \text{ isn't prime either}\\ \text{so that can't be a general rule regarding the primality of powers of two plus or minus 1}$$

$$\text{you just need to prove that the two numbers are coprime, i.e. have no common factors}$$

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Feb 10, 2019
edited by Rom  Feb 10, 2019