3^n+1 has an odd prime divisor

[wsldam]

Prove that 3ⁿ + 1 has an odd prime divisor for all natural numbers > 1. I tried using order but it didn't really get me anywhere. Would prefer hints rather than complete solutions. Thanks.

 

[AlephZero]

If you can't see how to prove this sort of result, start by playing around with the numbers.

Work out the values of a few numbers, 3²+1, 3³+1, 3⁴+1, ... What do you notice about them?

 

[tiny-tim]

welcome to pf!

hi wsldam! welcome to pf!

Prove that 3ⁿ + 1 has an odd prime divisor for all natural numbers > 1.

isn't that another way of saying that 2^m can never be of the form 3ⁿ + 1 ?

 

[AlephZero]

hi wsldam! welcome to pf!


isn't that another way of saying that 2^m can never be of the form 3ⁿ + 1 ?

It is also saying that 3ⁿ + 1 is never an odd prime (but that's easy to show).

 

[Norwegian]

solution 1: Do it modulo 8
solution 2: Do it for m odd (mod 3) and m even (factor 2^m-1).

 

[tiny-tim]

nice!