From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: An equation related to FLT: Date: 12 Jan 1999 23:12:16 GMT Newsgroups: sci.math Keywords: Homogeneous ternary Diophantine equations like Fermat's >steiner@math.bgsu.edu (steiner) wrote, in part: > >>Let p be an odd prime. What(if anything) is known about the Diophantine equation > >>x^p+y^p= pz^p? John Savard wrote: >If p were 2 (I know, that isn't an odd prime) x^2 + y^2 = 2z^2 has the >trivial solution x=y=z. > >Although x^3 + y^3 can't be a cube, can it be 3 times a cube? Well, 1^3 + (-1)^3 = 3 * 0^3, and of course you can scale this example. But then we can use this point as the origin on what is now an elliptic curve (A243 in Cremona's tables) and deduce that it has rank 0 and no torsion -- in short, no other rational points. The general problem of solving x^3 + y^3 = k z^3 in integers has arisen before in this newsgroup; see index/14H52.html or specifically 97/2cubes Note that [1, -1, 0] is a solution to the proposed problem for all odd values of p. For any given p, there will be at most a finite number of solutions [x, y, z] up to scaling, since the equation describes a curve of genus greater than one; but that general result is not effective (we can't readily determine whether there are solutions for any particular p). Clearly for p an odd prime we must have x = -y mod p and z = 0 mod p. For p=4 there is no solution, since that would require x^4 + y^4 = (2 z^2)^2 and Fermat showed no two fourth powers can sum to a square. I don't know about the general case, but Henri Darmon and Andrew Granville discussed an even more general equation Ax^p+By^q=Cz^r in 1995 (Bull. London M.S.) Let me remark that this equation is not only very close to the Fermat equation but that the algebraic-number-theory proofs of cases of FLT seem to circle around just this issue of extra factors of p. I would really like to see a solution to this equation for some p -- that way, people who claim to have a proof of FLT would be invited to see how their proof compares to this example (they might easily be convinced their "proof" applies to this equation too); it would be fun to then reveal a counterexample. dave