Can Mathematical Induction Prove the Primality of 2n-1 and 2n+1?

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The discussion centers on proving that if either 2n-1 or 2n+1 is prime for n > 2, then the other must not be prime. Participants suggest starting with a base case, such as n = 3, where 2^3-1 equals 7 (prime) and 2^3+1 equals 9 (not prime). The approach involves using mathematical induction, assuming the statement holds for a particular n and then proving it for n + 1. There is also mention of specific conditions under which 2^n-1 and 2^n+1 can be prime, particularly focusing on odd and even values of n. The discussion emphasizes the need for a structured proof using induction to establish the relationship between the two expressions.
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Homework Statement



Prove that if one of the numbers 2n-1 and 2n+1 is prime, n>2, then the other number is not

Homework Equations





The Attempt at a Solution

 
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What have you tried?
 
I don't even know how to start.
 
Part 1: Pick one of the numbers, and assume it is a prime larger than 2. Then show that the other number is not prime.

Part 2: Now pick the other number, and assume it is a prime larger than 2. Then show that the other number is not prime.
 
I don't know...it the result is correct but...2^n-1 is prime when n is an odd number...not all odd number but n has to be of the odd form...and 2^n+1 is prime...when n is some even number...

can somebody tell me if it is correct...
 
Have you thought about using mathematical induction?

Set up your base case: n = 3
You will show that 2^3-1 = 8 - 1 = 7 is prime and 2^3 + 1 = 9 is not since 9 = 3 \cdot 3.

Assume that it's true for n. Then prove the case for n + 1.
 

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