proof of fermat-catalan conjecture through the ... - Nardelli - Xoom.it
proof of fermat-catalan conjecture through the ... - Nardelli - Xoom.it
proof of fermat-catalan conjecture through the ... - Nardelli - Xoom.it
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Versione 1.022/3/2013Pagina 7 di 4943 8 + 96222 3 = 30042907 2{rad (43*2*3*7*29*79*109*275623)} 62π> 902576261010649124.303.909.686.222 1,645 = 153259525699804607160993,02 > 902.576.261.010.6491 1 1+ + = 0,95833….m n kThe first <strong>of</strong> <strong>the</strong>se (1 m +2 3 =3 2 ) is <strong>the</strong> only solution where one <strong>of</strong> a, b or c is 1; this is <strong>the</strong> Catalan<strong>conjecture</strong>, proven in 2002 by Preda Mihailescu. Technically, this case leads infin<strong>it</strong>ely many solutions(since we can pick any m for m>6), but for <strong>the</strong> purposes <strong>of</strong> <strong>the</strong> statement <strong>of</strong> <strong>the</strong> Fermat-Catalan<strong>conjecture</strong> we count all <strong>the</strong>se solutions as one.PROOFa m + b n = c k1 1 1+ + < 1m n kAs <strong>the</strong> new abc <strong>conjecture</strong> is given:2π{rad (a m b n c k )} 6> c kLet’s applyrad (a m b n c k ) ≤ abc < c m kc n kc
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