Date: Tue, 06 Jan 1998 12:28:40 -0300 From: Dario Alejandro Alpern To: rusin@math.niu.edu Subject: Sum of two powers = Square David, I was reading your page located at: 97/ferm.2.3.3 It contains a reply by Gerry Myerson to the list I sent to USENET in Nov 20th, 1997. Notice that that list (that you can find at Dejanews) contained a lot of solutions that were not both cubes. Filtering the a^3 + b^3 = c^2 cases: 1 ^ n + 2 ^ 3 = 3 ^ 2 2 ^ 7 + 17 ^ 3 = 71 ^ 2 5 ^ 4 + 6 ^ 3 = 29 ^ 2 7 ^ 4 + 15 ^ 3 = 76 ^ 2 7 ^ 5 + 393 ^ 3 = 7792 ^ 2 10 ^ 5 + 41 ^ 3 = 411 ^ 2 17 ^ 4 + 42 ^ 3 = 397 ^ 2 17 ^ 7 + 76271 ^ 3 = 21063928 ^ 2 57 ^ 5 + 52684 ^ 3 = 12092581 ^ 2 97 ^ 4 + 3135 ^ 3 = 175784 ^ 2 185 ^ 5 + 5879 ^ 3 = 647992 ^ 2 307 ^ 4 + 2262 ^ 3 = 143027 ^ 2 383 ^ 4 + 25800 ^ 3 = 4146689 ^ 2 433 ^ 4 + 26462 ^ 3 = 4308693 ^ 2 545 ^ 4 + 954 ^ 3 = 298483 ^ 2 804 ^ 4 + 1417 ^ 3 = 648613 ^ 2 1727 ^ 4 + 34544 ^ 3 = 7079295 ^ 2 1849 ^ 4 + 96222 ^ 3 = 30042907 ^ 2 (*) 2316 ^ 4 + 34177 ^ 3 = 8288063 ^ 2 2981 ^ 4 + 4959 ^ 3 = 8893220 ^ 2 4446 ^ 4 + 45457 ^ 3 = 22015007 ^ 2 7189 ^ 4 + 63855 ^ 3 = 54142096 ^ 2 9959 ^ 4 + 81840 ^ 3 = 101907569 ^ 2 Notice that 1849=43^2, so (*) has the form a^8 + b^3 = c^2 There are some solutions here that do not appear in your page located at: 96/bignum.exmp Have a nice day! -- Dario Alejandro Alpern Buenos Aires - Argentina http://members.tripod.com/~alpertron (en castellano) http://members.tripod.com/~alpertron/ENGLISH.HTM (English) Si su navegador no soporta JavaScript: http://members.tripod.com/~alpertron/INDEX2.HTM If your browser does not support JavaScript: http://members.tripod.com/~alpertron/ENGLISH2.HTM Antes era fanfarron... Ahora soy perfecto!! ============================================================================== Date: Thu, 08 Jan 1998 15:54:05 -0300 From: Dario Alejandro Alpern To: rusin@math.niu.edu Subject: Sum of two powers = Square Newsgroups: sci.math.num-analysis,sci.math These are the sums that I found using the UBASIC program listed in my post of Nov 20th, 1997 (search for it in http://www.dejanews.com): 1 ^ n + 2 ^ 3 = 3 ^ 2 5 ^ 4 + 6 ^ 3 = 29 ^ 2 7 ^ 4 + 15 ^ 3 = 76 ^ 2 17 ^ 4 + 42 ^ 3 = 397 ^ 2 97 ^ 4 + 3135 ^ 3 = 175784 ^ 2 307 ^ 4 + 2262 ^ 3 = 143027 ^ 2 383 ^ 4 + 25800 ^ 3 = 4146689 ^ 2 433 ^ 4 + 26462 ^ 3 = 4308693 ^ 2 545 ^ 4 + 954 ^ 3 = 298483 ^ 2 804 ^ 4 + 1417 ^ 3 = 648613 ^ 2 1351 ^ 4 + 608399 ^ 3 = 474554340 ^ 2 1727 ^ 4 + 34544 ^ 3 = 7079295 ^ 2 1849 ^ 4 + 96222 ^ 3 = 30042907 ^ 2 (1) 2316 ^ 4 + 34177 ^ 3 = 8288063 ^ 2 2981 ^ 4 + 4959 ^ 3 = 8893220 ^ 2 4446 ^ 4 + 45457 ^ 3 = 22015007 ^ 2 7189 ^ 4 + 63855 ^ 3 = 54142096 ^ 2 7321 ^ 4 + 208464 ^ 3 = 109233295 ^ 2 9847 ^ 4 + 573242 ^ 3 = 444716637 ^ 2 9959 ^ 4 + 81840 ^ 3 = 101907569 ^ 2 17514 ^ 4 + 931177 ^ 3 = 949475843 ^ 2 19110 ^ 4 + 285769 ^ 3 = 395856397 ^ 2 22247 ^ 4 + 568920 ^ 3 = 655054991 ^ 2 23783 ^ 4 + 249102 ^ 3 = 579133627 ^ 2 32039 ^ 4 + 273695 ^ 3 = 1036435896 ^ 2 44113 ^ 4 + 452862 ^ 3 = 1969675733 ^ 2 78895 ^ 4 + 680826 ^ 3 = 6249719699 ^ 2 7 ^ 5 + 393 ^ 3 = 7792 ^ 2 10 ^ 5 + 41 ^ 3 = 411 ^ 2 57 ^ 5 + 52684 ^ 3 = 12092581 ^ 2 185 ^ 5 + 5879 ^ 3 = 647992 ^ 2 3 ^ 5 + 11 ^ 4 = 122 ^ 2 2 ^ 7 + 17 ^ 3 = 71 ^ 2 17 ^ 7 + 76271 ^ 3 = 21063928 ^ 2 43 ^ 8 + 96222 ^ 3 = 30042907 ^ 2 (1) The equations marked with (1) are, of course, the same. Notice that the gcd of the three bases are 1. An open question: The set of a^4 + b^3 = c^2 with gcd(a,b,c)=1 is finite or infinite? -- Dario Alejandro Alpern Buenos Aires - Argentina http://members.tripod.com/~alpertron (en castellano) http://members.tripod.com/~alpertron/ENGLISH.HTM (English) Si su navegador no soporta JavaScript: http://members.tripod.com/~alpertron/INDEX2.HTM If your browser does not support JavaScript: http://members.tripod.com/~alpertron/ENGLISH2.HTM Antes era fanfarron... Ahora soy perfecto!! ============================================================================== Date: Fri, 9 Jan 1998 19:06:34 -0600 (CST) From: Dave Rusin To: alpertron@hotmail.com Subject: Re: Sum of two powers = Square >An open question: The set of a^4 + b^3 = c^2 with gcd(a,b,c)=1 is finite >or infinite? I believe it has been shown that a^m+b^n=c^k has infinitely many solutions if 1/m+1/n+1/k > 1 finitely many solutions if 1/m+1/n+1/k < 1 The cases 1/m+1/n+1/k = 1 are limited to (m,n,k) = (2, 3, 6), (2, 4, 4), (3, 3, 3), the last two being Fermat's theorem (proving FLT for n=4) and Euler's theorem (i.e. FLT for n=3). There is some deep connection to the "Coxeter" groups defined by < x, y, z | x^m=y^n=z^k=1 > and their action (in groups generated by reflections) on the plane; the condition 1/m+1/n+1/k > 1 is equivalent to finiteness of the group. I will probably include your email on my site in some appropriate place -- if that's OK with you? dave ============================================================================== Date: Mon, 12 Jan 1998 09:20:37 -0300 From: Dario Alejandro Alpern To: Dave Rusin Subject: Re: Sum of two powers = Square Dave Rusin wrote: > >An open question: The set of a^4 + b^3 = c^2 with gcd(a,b,c)=1 is finite > >or infinite? > > I believe it has been shown that a^m+b^n=c^k has > infinitely many solutions if 1/m+1/n+1/k > 1 > finitely many solutions if 1/m+1/n+1/k < 1 > > The cases 1/m+1/n+1/k = 1 are limited to (m,n,k) = (2, 3, 6), (2, 4, 4), > (3, 3, 3), the last two being Fermat's theorem (proving FLT for n=4) and > Euler's theorem (i.e. FLT for n=3). > > There is some deep connection to the "Coxeter" groups defined by > < x, y, z | x^m=y^n=z^k=1 > > and their action (in groups generated by reflections) on the plane; the > condition 1/m+1/n+1/k > 1 is equivalent to finiteness of the group. > > I will probably include your email on my site in some appropriate place -- > if that's OK with you? > Of course it is OK, thanks. > dave I have included in my web page (see addresses below) all the sums of powers I found. Even though the forms a^4 + b^3 = c^2 and a^5 + b^3 = c^2 have infinite solutions (as you noticed above) I could not find a method to find all solutions. In sci.math some people kindly gave me several sets of infinite solutions to these equations, but not all. Then, I am making a table of the first solutions to these equations. Have a nice day! -- Dario Alejandro Alpern Buenos Aires - Argentina http://members.tripod.com/~alpertron (en castellano) http://members.tripod.com/~alpertron/ENGLISH.HTM (English) Si su navegador no soporta JavaScript: http://members.tripod.com/~alpertron/INDEX2.HTM If your browser does not support JavaScript: http://members.tripod.com/~alpertron/ENGLISH2.HTM Antes era fanfarron... Ahora soy perfecto!!