We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.

+0

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

0
210
1
+159

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

### 1+0 Answers

#1
+6045
+1

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

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

.
Feb 10, 2019
edited by Rom  Feb 10, 2019