From: nepalese@my-deja.com Subject: Re: a triangle question Date: Sun, 12 Sep 1999 23:10:15 GMT Newsgroups: sci.math Keywords: Triangles with integer sides and area, two sides =consecutive squares In article <7rc0t5$84m$1@gannett.math.niu.edu>, rusin@vesuvius.math.niu.edu (Dave Rusin) wrote: > In article <7rbge7$mel$1@nnrp1.deja.com>, wrote: > >Given a triangle with sides [x^2, (x+1)^2, y] there exist infinitely > >many solutions where x and y are integers giving the area also as an > >integer. > > How do you know there are infinitely many solutions? > I mean, is that a conjecture you've made based on experimental data, > or is this a homework problem you need help with? > > >I have been able to generate some solutions, i.e., 16,25,39 and > >25,36,29. I want to prove this statement. I have tried the Heron's > >formula so far. Any help is much appreciated. > > OK, I'll give you some more data; perhaps these will be sufficient for > you to discover that there are infinitely many solutions of at least > one type, without giving away the whole answer: [x^2, (x+1)^2, y]= > [16, 25, 39] > [625, 676, 1299] > [21904, 22201, 44103] > [748225, 749956, 1498179] > > Nice problem, actually. > > dave > I suppose we can generate more answers that will satisfy the given. How can I prove the statement though. It is more like a hw problem that one of my colleague asked me to help with. Nepalese > Sent via Deja.com http://www.deja.com/ Share what you know. Learn what you don't. ============================================================================== From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: a triangle question Date: 14 Sep 1999 00:23:09 GMT Newsgroups: sci.math In article <7rbge7$mel$1@nnrp1.deja.com>, wrote: >Given a triangle with sides [x^2, (x+1)^2, y] there exist infinitely >many solutions where x and y are integers giving the area also as an >integer. In article <7rc0t5$84m$1@gannett.math.niu.edu>, rusin@vesuvius.math.niu.edu (Dave Rusin) wrote: > OK, I'll give you some more data; perhaps these will be sufficient for > you to discover that there are infinitely many solutions of at least > one type, without giving away the whole answer: [x^2, (x+1)^2, y]= > [16, 25, 39] > [625, 676, 1299] > [21904, 22201, 44103] > [748225, 749956, 1498179] In article <7rhbse$h9v$1@nnrp1.deja.com>, wrote: >I suppose we can generate more answers that will satisfy the given. How >can I prove the statement though. It is more like a hw problem that one >of my colleague asked me to help with. I was trying to suggest a set of solutions which happen to satisfy another "obvious" relationship. Discover it, then discover the condition on x which implies that the integer y which satisfies this relationship will yield a triangle with integer area. Then show that there are infinitely many integers x meeting this condition. dave ============================================================================== [Answer: the surface in question is defined by the equation of Heron's formula: -(2*x+1+y)*(2*x+1-y)*(2*x^2+2*x+1-y)*(2*x^2+2*x+1+y) must be square. Clearly there is a sign problem if y is large relative to x, so we try taking y in the neighborhood of what is necessary to keep one factor constant -- or as is suggested by the example, we try y = x^2 + (x+1)^2 - 2 . With this substitution, we find we need only select x to make 2 x^2 + 2 x - 4 a square, which is a Pell's equation, and so we may find infinitely many integer solutions. I don't know the full set of rational or integer points on this elliptic surface. --djr]