Proof that No two Fermat numbers have a common divisor greater than 1

 

$$\begin{array}{l}Supposethat{F}_{n}and{F}_{n+k},where,aretwoFermatnumbers,andthat\\ \\ m|F_n,m|F_{n+k}\\ \\ \frac{F_{n+k}-2}{F_n}=\frac{2^{2^{n+k}}-1}{2^{2^n}+1}=\frac{x^{2^k}-1}{x+1}=x^{2^k-1}-x^{2^k-2}+\dots -1\\ \\ Andso{F}_n|F_{n+k}-2,whichimpliesthatm|F_{n+k}-2.\\ \\ Hence\\ \\ m|F_{n+k}andm|F_{n+k}-2;\\ \\ Andthereforem|2.Since{F}_{n}isodd,m=1,whichprovesthetheorem.\end{array}$$