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

Versione 1.022/3/2013Pagina 17 di 492. PROOF OF THE CATALAN'S CONJECTURECatalan's conjecture (or Mihăilescu's theorem) is now a theorem that was conjectured by themathematician Eugene Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu.2 3 and 3 2 are two powers of natural numbers, whose values 8 and 9 respectively are consecutive. Thetheorem states that this is the only case of two consecutive powers. That is to say, that the onlysolution in the natural numbers offor c, k, b, n > 1c k − b n = 1is1 + b n = c k1 + 2 3 = 3 2so worth the inequality1 1 1+ + < 1m n kBut what value we have to choose for m?We use the trick of taking m = ∞ so that we have he following inequality1 ∞ + 2 3 = 3 21 1 + < 1n kLet’s apply the new abc conjecture and so let's show this conjecture:rad (1*b n c k ) ≤ bc < c n kc