proof of fermat-catalan conjecture through the ... - Nardelli - Xoom.it
proof of fermat-catalan conjecture through the ... - Nardelli - Xoom.it
proof of fermat-catalan conjecture through the ... - Nardelli - Xoom.it
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Versione 1.022/3/2013Pagina 48 di 49This is sometimes known as <strong>the</strong> Goldbach-Euler <strong>the</strong>orem.But what happens if we add <strong>the</strong> differences <strong>of</strong> <strong>the</strong> reciprocals <strong>of</strong> <strong>the</strong> perfect powers?∑ ∞ 1= p k +− pk2 1k= 41 + 1 + 71 + 91 + 21 + 51 + 41 + 131 +….= ∞Since <strong>it</strong> must appear all natural numbers (w<strong>it</strong>h also duplicates) we have a sum a b<strong>it</strong> more <strong>of</strong> <strong>the</strong> series<strong>of</strong> <strong>the</strong> Harmonic series, that’s a divergent infin<strong>it</strong>e series.This proves also that <strong>the</strong> gaps in <strong>the</strong> sequence <strong>of</strong> perfect powers tend to infin<strong>it</strong>y.
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