Diophantine EquationsDate: 06/29/2001 at 13:45:36 From: Deva Mishra Subject: Diophantine Equations I have recently become interested in number theory. I am reading a book by Andrew Weil that discusses Diophantine Equations, and he proposed a question: Find rational x and y such that x^2+x^2*y^2 and y^2+x^2*y^2 are perfect squares. Or, more simply, x^2+x^2*y^2 = m^2 and y^2+x^2*y^2 = n^2, where n and m are rational numbers. I tried applying the typical form to the two equations, but I got stuck. The solution would be greatly appreciated. Thank you very much. Deva Mishra Date: 06/29/2001 at 16:10:39 From: Doctor Jaffee Subject: Re: Diophantine Equations Hi Deva, Here is how I would approach this problem. x^2+x^2*y^2 = x^2(1 + y^2) Since x^2 is a perfect square for any rational value of x, we need to concentrate on 1 + y^2. Consider any Pythagorean triple (a,b,c) and let y = a/b. a^2 + b^2 c^2 Then 1 + y^2 = 1 + a^2/b^2 = ------------------ = ------, so m = c/b b^2 b^2 Likewise, you can find a value for x. Besides Pythagorean triples, you can let x or y = 0. Also, integers such as -a and -b can be used where (a,b,c) is a Pythagorean triple. I hope my answer has helped you understand the problem better. If you want to discuss it more, or if you have other questions, write back. Thanks for writing to Ask Dr. Math. - Doctor Jaffee, The Math Forum http://mathforum.org/dr.math/ |
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