From: snowe@rain.org (Nick Halloway) Newsgroups: sci.math Subject: Re: X^a + Y^b = Z^c Date: 28 Aug 1996 22:21:45 GMT Bill Center writes: >Conjecture: >X^a + Y^b = Z^c has no solutions for coprime X,Y,Z when each exponent >a,b,c > 2. If you find a counter example, PLEASE let me know. I asked someone who proved a related result; here is his e-mail quoted by permission. There are no known counterexamples to this conjecture and Henri Darmon and Andrew Granville proved that there can only be finitely many for given exponents a,b,c. e-mail from Andrew Granville: ------------------------------------------------------------------ There are no known examples of coprime integers x,y,z and a,b,c>2 with x^a + y^b = z^c. I do think that the question is a little `ad hoc' in the sense that if there are examples, there are almost certainly finitely many, and so it's a question of whether or not an extraordinary accident happens. Note that the `correct' conjecture is that if r,s,t are positive integers then there are only finitely many (x,y,z,a,b,c) of positive integers, with x,y,z pairwise coprime and 1/a + 1/b + 1/c < 1 and r x^a + s y^b = t z^c This follows from the `abc-conjecture', and was even proved, when a,b,c are fixed, by Darmon and me in the paper you referred to. Why I `object' to the question with a,b,c>2, is that if we take r=s=1, t=9 then we can obviously find the solution (1,2,1,3,3,3) above; but still there are only finitely many solutions. It is thus an `accident' of the coefficients whether or not there is some isolated example, and this probably sheds little light on other issues. However I do think that it is worth trying a `clever' search for examples. The examples (with one exponent 2) by Beukers and Zagier, which you will find quoted in our paper, show the danger of assuming that just because there are no more small examples, that you have exhausted all solutions. So good luck if you have the computer cycles and determination! Our paper is called `On the equations z^m = F(x,y) and Ax^p+By^q=Cz^r, and may be found in the Bulletin of the London Mathematical Society 27 (1995), 513--543. Sincerely, Andrew Granville -------------------------------------------------------- There are however counterexamples with one exponent = 2 which were posted by Andrew Granville on an old post to the nmbrthry mailing list: 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. ============================================================================== From: snowe@rain.org (Nick Halloway) Newsgroups: sci.math Subject: Re: X^a + Y^b = Z^c Date: 29 Aug 1996 01:09:40 GMT Nick Halloway (snowe@rain.org) wrote: : There are however counterexamples with one exponent = 2 which were : posted by Andrew Granville on an old post to the nmbrthry : mailing list: : : 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, \ \ ^^^ should be 43^8 : 33^8 + 1549034^2 = 15613^3. ============================================================================== From: alf@mpce.mq.edu.au (Alf van der Poorten) Newsgroups: sci.math Subject: x^a + y^b = z^c Date: Thu, 29 Aug 1996 09:52:10 +1000 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. -----------------------