Clausen Factorization For The sixth Fermat number

Published On :2021-06-15 15:49:00

 

begin{array}{l}\ {F_6} = {2^{64}} + 1 = {2^{48}} * {2^{16}} + 1\\ = 281474976710656 * {2^{16}} + 1\\ = 281473950092895 * {2^{16}} + 1026617761 * {2^{16}} + 1\\ = 1071 * 262814145745 * {2^{16}} + 2628141456816 * {2^8} + 1\\ = 1071 * 262814145745 * {2^{16}} + left( {1071 + ;262814145745} right) * {2^8} + 1\\ = left( {1071 * ;{2^8} + 1} right)left( {262814145745 * {2^8} + 1} right)\\ = 274177 * 67280421310721.\ end{array}