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

Versione 1.022/3/2013Pagina 44 di 49Now we see the same table for the conjecture Pillai, limited to the exponents 3 and 4 in place of 2 and3.x^3 y^4 Algebraicdifferencey^4- x^3Square root ofthe difference3^3 = 27 2^4 = 16 11 3,31 14^3=64 3^4=81 17 4,12 75^3=125 4^4= 256 131 11,44 16^3= 216 5^4=625 409 20,22 97^3=343 6^4=1296 953 30,87 38^3=512 7^4=2401 1889 43,46 99^3= 729 8^4=4096 3367 58,02 710^3=1000 9^4=6561 5561 74,57 111^3=1331 10^4= 10000 8669 93,10 912^3=1728 11^4= 14641 12913 113,63 313^3=2197 12^4=20736 18539 136,15 914^3=2744 13^4=28561 25817 160,67 715^3=3375 14^4=38416 35041 187,19 1… … … … …Observationsfinal digits1, 3, 7 and 9 ofthe differencesNow, however, we note that the differences are still growing faster than in the table above, andtherefore also their square roots.Now the differences between the final digits there is also the number 3, but there is a little more orderin the sequences of the final digits: we identify two identical successive groups: 7 1 9 3 9:07 1 9 3 9after initial sequence 1; but also the third group begins with 7 and 1.Further calculations with successive powers and their differences (from Pillai's conjecture) show someinteresting regularities, which we quote:Consecutive powers (for conjecture Pillai)1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81,100,121,125,128,144, 169,196,216, 225,243, 256, 289, 324,343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000,1024, 1089, 1156,1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764