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

Versione 1.022/3/2013Pagina 42 di 496. EXAMPLES OF PILLAI’S CONJECTUREWe see a table with x and y consecutive numbers with exponents 2 and 3, and then a similar table withexponents 3 and 4 for Pillaix^2 y^3 Algebraicdifferencey^3-x^2Square root ofthe difference3^2 = 9 2^3=8 1 1 14^2=16 3^3=27 11 3,31 15^2= 25 4^3= 64 39 6,24 96^2=36 5^3=125 89 9,43 97^2=49 6^3= 216 167 12,92 78^2=64 7^3=343 279 16,70 99^2=81 8^3=512 431 20,70 110^2=100 9^3=729 629 25,07 911^2=121 10^3=1000 879 29,64 912^2=144 11^3=1331 1187 34,45 713^2=169 12^3=1728 1559 39,48 914^2=196 13^3=2197 2001 44,73 115^2=225 14^3=2744 2519 50,18 9… … … … …Observationsfinal digits1, 7, 9 of thedifferencesAs we can see, the successive differences grow more and tend to infinity (their square roots growabout 4 or 5 at least up to x = 15), and therefore the only possible solution is the first,with one difference, as Catalan's conjecture. Furthermore, we note that the final digits of thedifferences are always 1, 7 and 9, are identified two successive groups of 9, 9, 7, 9, 1 after the pair 1, 7Initial.They are absent then the final digits 3 and 5.The fact that he chose:x 3 - (x + 1) 2 = kx 3 - x 2 - 2x -k - 1 = 0The solutions of this equation gives the values of x, being the divisors of the number (k – 1), by therule of Ruffini.