From: bbrock@pepperdine.edu (Bradley Brock) Newsgroups: sci.math.numberthy Subject: Re: x^a + y^b = z^c Date: 29 Aug 96 12:27:04 GMT > Conjecture: > > X^a + Y^b = Z^c has no solutions for coprime naturals X,Y,Z when each > exponent a,b,c > 2. To answer this question one need only quote Granville's post to this list from 2 1/2 years ago. The bottom line is that there are no known solutions and that for fixed a, b, & c, there can be at most finitely many. Brad -- From: Andrew Granville Subject: x^p+y^q=z^r Comments: To: NmbrThry@vm1.nodak.edu To: Multiple recipients of list NMBRTHRY In reply to a question of Dave Davis on x^p+y^q=z^r: Henri Darmon and I have been looking at integer solutions to the above equations. It is trivial to find lots for many different p,q,r. For example, the following parametric solution exists for exponents p,p,p+1: (ac)^p+(bc)^p = c^(p+1) where c = a^p+b^p. To rid us of such upstart solutions, let us restrict ourselves to solutions where x and y are coprime -- what we will call `proper solutions'. It is `well-known'(see Dickson) that 1) The above equation has inf many proper solns whenever 1/p+1/q+1/r>1. 2) The above equation has no proper solns whenever 1/p+1/q+1/r=1 except 3^2-2^3=1^6 What Darmon and I have proved is: If 1/p+1/q+1/r < 1 then there are only finitely many proper solutions to x^p+y^q=z^r. The main tool we use is Faltings' theorem (nee Mordell's conjecture); which we apply via an appropriate descent argument. A preprint will soon be available (send email to me for this preprint). Small examples of proper solutions to x^p+y^q=z^r with 1/p+1/q+1/r < 1 are: 1+2^3=3^2, \ \ 2^5+7^2=3^4, \ \ 7^3 + 13^2 = 2^9, \ \ 2^7+17^3=71^2, \ \ 3^5+11^4=122^2 Extraordinarily large solutions have been found recently by Beukers and Zagier: 17^7 + 76271^3 = 21063928^2, \ \ 1414^3 + 2213459^2 = 65^7, \ \ 9262^3 + 15312283^2 = 113^7, \ \ 3^8 + 96222^3 = 30042907^2, \ \ 33^8 + 1549034^2 = 15613^3. It amazes me that there is such a gap between the two sets of solutions. It may be that these are all of the solutions to the above, but I thought that before Beukers and Zagier got involved, so maybe I am wrong again. Can anyone compute some new examples ? That would be very interesting. Andrew Granville andrew@sophie.math.uga.edu ============================================================================== From: alf@mpce.mq.edu.au (Alf van der Poorten) Newsgroups: sci.math.numberthy Subject: Re: x^a + y^b = z^c Date: 29 Aug 96 12:27:07 GMT Nick Halloway asks about Subject: x^a + y^b = z^c The following conjecture came up in sci.math. A quick computer search found no counterexamples, although there are plenty of counterexamples for X,Y, Z not coprime. Does anyone know of a counterexample? If you do posting it in sci.math would be appreciated. Conjecture: X^a + Y^b = Z^c has no solutions for coprime naturals X,Y,Z when each exponent a,b,c > 2. ---------- This conjecture, which I call the Generalized Fermat Conjecture in my book `Notes on Fermat's Last Theorem' (Wiley--Interscience, 1996), surfaced in the context of Henri Darmon and Andrew Granville, `On the equation $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$', to appear, or recently appeared, in {\it Bull.\ London Math.\ Soc.\/}). It's easy to concoct seemingly nontrivial solutions with $X$, $Y$ and $Z$ sharing a common factor; so they're coprime from hereon: Even then there are paramtrised solutions (so infinitely many) if $1/a+1/b+1/c>1$, but then of course at least one of the exponents is $2$. There are only finitely many solutions if $1/a+1/b+1/c\le1$. There are $10$ such interesting solutions known: It's not too hard to notice the solutions $13^2+7^3=2^9$, $2^7+17^3=71^2$, $2^5+7^2=3^4$ and $3^5+11^4=122^2$. I suppose one might include $1+2^3=3^2$, if only out of respect for history, for it provides the only known solution to Catalan's problem of finding all solutions to $z^t-y^s=1$. We'll deem that $1=1^7$, say. One's computer will probably get tired before finding any solutions beyond these five. Yet five more solutions are now known (by courtesy of the computers of Frits Beukers and of Don Zagier), a severe blow for the Law of Small Numbers: $17^7+76271^3&=21063928^2$, $1414^3+2213459^2&=65^7$, $33^8+1549034^2&=15613^3$, $9262^3+15312283^2&=113^7$, $43^8+96222^3&=30042907^2$. Of course all these cases have an exponent $2$, so the GFC may well be true. ------------------------ Alf van der Poorten ceNTRe for Number Theory Research alf@mpce.mq.edu.au http://www.mpce.mq.edu.au/~alf/ fax: +61 2 9850 9502 voice: +61 2 9850 9500 home: +61 2 9416 6026 !!All Sydney Telephone Numbers changed 29/07/96: the 9 is extra Do you enjoy mathematics? Rush off to order a copy of "Notes on Fermat's Last Theorem" published by Wiley-Interscience (US$44.95 but! AU$90 rrp). It's just AU$58+pp from DA: mailto://service@dadirect.com.au. For details see http://www.mpce.mq.edu.au/~alf/NotesonFLT.html. "The poetry far excels that normally found in math books". H W Lenstra. "I love the book. Thanks for writing it. If you're ever in the Cotswolds come and stay". K B MD "... it should be bedtime reading for every mathematician". Ram Murty "Notes on FLT" will be published in Japanese by Morikita Shuppan, Tokyo. -----------------------