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

Versione 1.022/3/2013Pagina 19 di 49⎛ 1 1 ⎞⎜ + ⎟⎝ n k ⎠6≤ 2π= 0,6079…we cannot have integer solutions.We can prove this way:If n = 3 we have1 + b 3 = c khence we have solutions if and only if61 1⎝ n2π = 0,6079… < ⎛ ⎞⎜ + ⎟ k ⎠< 161 1+⎝ k2π = 0,6079… < ⎛ ⎞⎜ ⎟3 ⎠< 123πk < 218 − π= 3,6417…k > 23= 1,5and hence1, 5 < k < 3,6417…Having to take only integer valuesk = 2 ork = 3 ,this is impossible otherwise1 + b 3 = c 3Has no sense, the difference between two cubes is never equal to 1