proof of fermat-catalan conjecture through the ... - Nardelli - Xoom.it

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Versione 1.022/3/2013Pagina 8 di 49Please not that we cannot write, as is as in the case of Fermat's Great Theorem the inequalityrad (a n b n c n ) < c 3becauseit doesn't applya < c and b < c.The values of a, b and c may be any as long as obviously > 12πc k < c 6⎛⎜⎝kmk ⎞+ + 1⎟ n ⎠This implies that we have no solutions ifk ≥2π6⎛⎜⎝kmk ⎞+ + 1⎟n ⎠and hence dividing by k1 ≥2π6⎛ 1 1 1 ⎞⎜ + + ⎟⎝ m n k ⎠It depends on values of m, n and k⎛ 1 1 1⎜ + +⎝ m n kand hence we have solutions if and only if6⎞⎟⎠6≤ 2π2π = 0,6079… < m n k= 0,6079…1 1 1+ + < 1
Versione 1.022/3/2013Pagina 9 di 49This is a stronger inequality of the initial statement1 1 1+ + < 1m n kwhich then is wrong, because for⎛ 1 1 1⎜ + +⎝ m n k⎞⎟⎠6≤ 2πwe cannot have integer solutions-and this is the lower limit.If1 1 1+ + > 1m n k= 0,6079…We have infinite solutions of one of (2, 2, k), (2, 3, 3), (2, 3, 4), (2, 3, 5).If1 1 1+ + = 1m n kWe have finitely many solutions of one of (2, 4, 4), (2, 3, 6), (3, 3, 3).In fact from the examples above we have seen that the MINIMUM value for⎛ 1 1 1 ⎞⎜ + + ⎟⎝ m n k ⎠was13 2 + 7 3 = 2 9
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