From: nikl@mathematik.tu-muenchen.de (Gerhard Niklasch) Newsgroups: sci.math Subject: Re: That "3 prime factors" problem Date: 24 Mar 1998 17:26:44 GMT Keywords: S-unit equations in Z In article <6f8ml6\$1f0\$1@news1.rmi.net>, Kurt Foster reduced Another Problem to that of solving the following diophantine equations: |> (A1) |2^C * 5^D - 3^B| = 1 |> |> (A2) |2^A * 3^B - 5^D| = 1 |> |> (A3) |3^B - 5^D| = 2 |> (B1) |2^A * 3^B - 5^D| = 1 |> |> (B2) |2^C * 5^D - 3^B| = 1 |> (C1) |2^A * 3^C -5^D| = 1 |> |> (C2) |2^A - 3^C * 5^D| = 1 where, for the purpose at hand, it would be sufficient to know that each of them has only finitely many solutions. Now it is well known that the `S-unit equation' X + Y = Z in integers X,Y,Z such that all prime divisors of XYZ come from a given finite set S of primes has only finitely many solutions in which X,Y,Z are coprime, and the solutions can be effectively determined. While this requires transcendental methods in general, L.J.Alex [Comm. Algebra 4 (1976), 77--100] arrived at the complete solution for S={2,3,5,7} by elementary methods. If you want to go a little beyond these primes, B.M.M.de Weger [Algorithms for Diophantine Equations, CWI Tract 65, 1989, Chapter 6: Thm.6.3 and Table II] gives the full solution set for S={2,3,5,7,11,13}. For S={2,3,5}, the complete solution set, up to playing games with signs and permuting summands, consists of 1 + 1 = 2 2 + 1 = 3 3 + 1 = 2^2 3 + 2 = 5 2^2 + 1 = 5 5 + 1 = 2*3 5 + 3 = 2^3 5 + 2^2 = 3^2 2^3 + 1 = 3^2 3^2 + 1 = 2*5 3*5 + 1 = 2^4 2^4 + 3^2 = 5^2 2^3*3 + 1 = 5^2 5^2 + 2 = 3^3 3^3 + 5 = 2^5 2^4*5 + 1 = 3^4 5^3 + 3 = 2^7 The application to the previous poster's exposition is left as an exercise. :^) (Gotta love this -- didn't take me long to hack a little GP/PARI loop to spit these out, starting from the bounds in de Weger's theorem since I don't have the other paper within arm's reach, _and_ to spot Yet Another Trivial Little Buglet in pari-gp-2.0.7.alpha whilst doing so...) Enjoy, Gerhard -- * Gerhard Niklasch * spam totally unwelcome * http://hasse.mathematik.tu-muenchen.de/~nikl/ ******* all browsers welcome * This .signature now fits into 3 lines and 77 columns * newsreaders welcome