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

Versione 1.022/3/2013Pagina 46 di 4967 prime69 = 3*2371 prime35 =5*738 = 2*1975 = 3*5*577 =7*779 prime81 =9^2= 3^447 prime36 =6^2We note that 4 occurs three times, twice on 3, 9 twice, 7 and 13 once, 6, 8, 10 and 12 zero times.The odd numbers in blue differ by 2 to groups of two, three or four consecutive numbers, with a groupof eight consecutive odd numbers. After this cycle falls to about half last number in blue (ex. 71/35 =2.02, 81/47 = 1.7) and after two or three sizes too small recurs again, obviously with consecutivenumbers gradually larger.They are present Fibonacci numbers (3, 5, 13, 21), square (4, 9, 16, 25, 36, 49, 81 and even a cube(27). We assume that this trend continues as the semiregular differences growth of consecutive powers,and could contribute to another definitive proof of Pillai's conjecture, according to which thesedifferences are repeated in a finite number of times.This means that the smaller numbers of this estimate will never be more later, and therefore theirpresence exists in a finite number of times, just as says the conjecture, so that proves true even if onlyby empirical means these numerical evidence, but not analytical.ConclusionsRemember that Pillai's conjecture states that for any given positive integer A there are only a finitenumber of pairs of perfect powers whose difference is A, but from the proof above that we have thatfor fixed positive integers A, B, C the equationCc k- Bb n = Ahas only finitely many solutions (b,c,n,k) with (n,k) ≠ (2,2).This is equivalent to saying that each positive integer A is valid and occurs only finitely many timesas a difference of perfect powers.