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

Versione 1.022/3/2013Pagina 35 di 495. PROOF OF TIJDEMAN'S CONJECTURETijdeman's conjecture states that there are at most a finite number of consecutive powers.Stated another way, the set of solutions in integers b, c, n, k of the exponential diophantine equationfor exponents n and k ≥ 2, is finite.A + b n = c kIf A =1 the theorem was proven by Mihailescu and it's the Catalan's conjecture, as we have seen beforeand is also known as Tijdeman's theorem.But with A ≥ 2, n and k ≥ 2 we have an unsolved problem, called the generalized Tijdeman problem. Itis conjectured that this set also will be finite. This would follow from a yet stronger conjecture of Pillaistating that the equationA + Bb n = Cc konly has a finite number of solutions, that we have shown above.In factLet’s apply the new abc conjecturerad (A*b n c k ) ≤ Abc < Ac n kcc k < (A) 62 2π πc 6⎛ k ⎞⎜ + 1⎟ ⎝ n ⎠Now to solve this inequality we assume that2πA 6= cπ2e6