- #1

cragar

- 2,552

- 3

## Homework Statement

Show that all fermat numbers are relatively prime.

## The Attempt at a Solution

If they share common factors then it should divide their difference so I look at

n>x

[itex] (2^{2^n}+1)-(2^{2^x}+1)=2^{2^x}(2^{2^n-2^x}-1) [/itex]

Now since fermat numbers are odd their factors will have to be contained in

[itex] (2^{2^n-2^x}-1) [/itex] Now if they share common factors then it should divide

[itex] (2^{2^n-2^x}-1)+(2^{2^n}+1)= (2^{2^n-2^x}+2^{2^n})=2^{2^n}(2^{-2^x}+1)[/itex]

Since fermat numbers are odd their factors must divide into

[itex] (2^{-2^x}+1) [/itex] but this is not an integer so this contradicts our assumption that it will divide our sum, so fermat numbers are relatively prime.