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