From: steiner@bgnet0.bgsu.edu (Ray Steiner) Newsgroups: sci.math.research Subject: Re: Perfect Powers Date: Thu, 27 Aug 1998 13:00:32 -0500 In article <35E44750.95516589@SouthWind.Net>, rgwv@southwind.net wrote: > Et al, > > It is well known that only the perfect powers that differ by 1 > are the integers 8 & 9. Perfect powers are integers which are the > result of raising the natural numbers to any power greater than one. > See Neil J.A. Sloane, M3326 in Encyc. of Integer Sequences by Academic > Press, 1995. For a difference of tow, we have 25 & 27, 3 has 1 & 4 and > 125 & 128, 4 has 4 & 8 and 32 & 36, and a difference of 5 has 4 & 9 and > 27 & 32. Question, what is the first pair (not necessarily consecutive) > perfect power numbers which differ by 6, 14, 34, 42 and 50. Thank you > in advance. I've checked all perfect power numbers up to 2*10^10. > > Mathematically yours, > Robert G. Wilson v, > PhD ATP / CF&GI Eh, so sorry, but it is NOT "well-known" that the only powers that differ by 1 are 8 and 9. This is Catalan's conjecture. The problem has been shown to be equivalent to solving x^p- y^q=1, where p and q are odd primes satisfying 10^5 < p < 3.31*10^12, 10^6 < q < 4.13*10^17, assuming p < q. Question: How do we resolve the problem for the ranges of p and q listed here?? For more on Catalan's conjecture, see the book by Paulo Ribenboim CATALAN'S CONJECTURE Are 8 and 9 the only consecutive powers? Academic Press, New York, 1994. Incidentally, a famous mathematician once told me that the solution of x^p- y^q = 6 would lead to the solution of another famous problem. Could anyone tell me what he was referring to? Regards, Ray Steiner ============================================================================== From: gerry@mpce.mq.edu.au (Gerry Myerson) Newsgroups: sci.math Subject: Re: x^2-1=y^m Date: Thu, 11 Mar 1999 09:58:13 +1100 In article , blang@club-internet.fr (Bruno Langlois) wrote: > Who can solve the diophantine equation : > x^2-1=y^m (x,y,m > 1) Chao Ko, On the diophantine equation x^2 = y^n + 1, xy \ne 0, Scientia Sinica (Notes) 14 (1965) 457--460, MR 32 #1164. See also Mordell, Diopantine Equations, pp. 302--304. Gerry Myerson (gerry@mpce.mq.edu.au)