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

Versione 1.022/3/2013Pagina 25 di 49This implies that we have no solutions ifk ≥2π6⎛⎜⎝kmk ⎞+ + 1 + e⎟n ⎠and hence dividing by k1 ≥2π6⎛ 1 1 1 e ⎞⎜ + + + ⎟⎝ m n k k ⎠It depends on values of n, k and e⎛ 1 1 1 e⎜ + + +⎝ m n k k⎞⎟⎠6≤ 2π= 0,6079…and hence we have solutions if and only if61⎝ m1n1kek2π = 0,6079… < ⎛⎞⎜ + + + ⎟ ⎠< 1withe ≥ 061⎝ m1n1k2π = 0,6079… < ⎛ ⎞⎜ + + ⎟ ⎠< 12 2 2π π πB 6 C−16(A 6) = cπ2e6and we have the case of Fermat-Catalan conjecture with A = 1, B = 1 and C =1, in fact: