From: elkies@ramanujan.math.harvard.edu (Noam Elkies) Newsgroups: sci.math.research Subject: 1,3,8,120,etc. (Re: s.m.r. Problems for the next century) Date: 21 Apr 1997 04:33:21 GMT In article <5jcof2$4ni$1@newsflash.concordia.ca>, MCKAY john wrote: >In article <1sxxlAAvvzVzEwJt@weburbia.demon.co.uk> > Phil Gibbs writes: >>Do there exist 5 positive integers such that the >>product of any two is one less than a square? >>It is an old problem even known to Diophantus >>himself. >Has this not been solved? See a paper by Stephen Muir (et al?) using >Baker's (then new) approximation results. Try Quarterly J. Math. about >1967. >Four numbers with this property are 1,3,8,120. Muir, I believe, proves >THIS set cannot be augmented. Does this solve the problem? Alas no, because there are infinitely many quadruples. This problem, and the solution for {1,3,8,120,x}, was even reported by Martin Gardner in his Sci.Am. column several decades ago. Does it really go back to Diophantus? I thought Diophantus never required integer solutions, only rational ones. (Are there even any examples of rational quintuples?) --Noam D. Elkies (elkies@math.harvard:edu) Dept. of Mathematics, Harvard University ============================================================================== From: Philip Gibbs Newsgroups: sci.math Subject: Re: Problem Solvers Needed! Date: Sun, 17 May 1998 10:49:35 +0100 In article <6j9nug$7tj@basement.replay.com>, mathedman@usa.net writes > The set A={1,3,8,120} has the interesting property >that if a and b are distinct elements of A then ab+1 >is a perfect square. It is easy to find many such sets. >For example: {2,4,12,420}, {2,4,420,14280}, {1,8,4095,139128}, >{2,12,2380,233244}, etc. > Are there infinitely many such sets?? > Find such a set with 5 members; with 6 members. > Is there an infinite set with this property?? All solutions of the following diophantine equation in positive integers have the property you defined: a^2 + b^2 + c^2 + d^2 - 2ab - 2ac - 2ad - 2bc - 2bd - 2cd - 4abcd = 4 It is possible to construct all the solutions of this equation using a recursive tree method and there are infinitely many. It has been conjectured (by me) that all such sets of four positive integers are solutions of this equation but there is no proof as yet. If it is correct then there are no such sets of five positive integers, but since you did not say positive integers how about 0, 1, 3, 8, 120 ? See my web page for more about this problem: http://www.weburbia.com/pg/diophant.htm Free electronic book: "Event-Symmetric Space-Time" by P. Gibbs http://www.weburbia.com/press/esst.htm http://www.weburbia.demon.co.uk/press/esst.htm "The universe is made of stories, not of atoms" Muriel Rukeyser ============================================================================== From: Pierre Bornsztein Subject: Re: Number Theory Date: Sat, 02 Oct 1999 07:27:01 +0200 Newsgroups: sci.math To: Sangjeong Kim Sangjeong Kim wrote: > > a, b, c are natural numbers. > If (ab+1)(bc+1)(ca+1) is perfect square number, prove ab+1, bc+1 and > ca+1 are also perfect > square number. > > This is a problem which one of my student asked. But I have no idea. So > if you have any idea, > help me please. > Thank you in advance. > Sangjeong Kim You will find a solution of this problem in Mathematics Magazine,vol.71,n°1(February 1998), p 61-63. Pierre. ============================================================================== From: duje@math.hr (Andrej Dujella) Subject: rank=8, torsion group=Z/2Z*Z/2Z Date: 26 Apr 99 14:52:55 GMT Newsgroups: sci.math.numberthy I found three examples of elliptic cuves over Q with torsion group isomorphic to Z/2Z * Z/2Z and with rank = 8. This improves my previous examples with rank = 7 (see A. Dujella: Diophantine triples and construction of high-rank elliptic curves over Q with three non-trivial 2-torsion points, Rocky Mountain J. Math., to appear). These elliptic curves are y^2=x*[x+(b-a)(d-c)]*[x+(c-a)(d-b)], where (a,b,c,d)= (32/91, 60/91, 1878240/1324801, 15343900/12215287), (17/448, 2145/448, 23460/7, 2352/7921) and (559/1380, 252/115, 24264935/2979076, 16454108/1703535). The quadruples {a,b,c,d} are subsets of rational Diophantine sextuples (sets of six positive rationals such that the product of any two of them is one less than a square) discovered by Philip Gibbs (P. Gibbs: Some rational Diophantine sextuples, math.NT/9902081; P. Gibbs: A generalised Stern-Brocot tree from regular Diophantine quadruples, math.NT/9903035). The ranks are computed with John Cremona's program MWRANK. Andrej Dujella Department of Mathematics University of Zagreb Bijenicka cesta 30 10000 Zagreb CROATIA http://www.math.hr/~duje/