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

Published On :2021-06-16 23:42:00

 

[begin{array}{l} Suppose;that;{F_n};and;{F_{n + k}},{rm{ }}where,;are;two;Fermat;numbers,;and;that\\ 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}} + ldots - 1;\\ And;so;{F_n}|{F_{n + k}} - 2;,which;implies;that;m|{F_{n + k}} - 2.\\ Hence\\ m|{F_{n + k}};and;;;;m|{F_{n + k}} - 2;\\ And;therefore;;m|2.;Since;{F_n};is;odd,;;m = 1,;which;proves;the;theorem.\ end{array}]