Beyond Pythagorean Triples
Date: 11/20/2001 at 10:49:52 From: lynn meredith Subject: Pythagorean triples I am a pupil studying a topic that covers the Pythagorean triples. For the higher piece I need to see and prove whether you can use the equations: a^3 + b^3 + c^3 = d^3 a^4 + b^4 + c^4 = d^4 Could you perhaps show me how these work, how you obtain numbers, and how to prove they are correct? Thank you!
Date: 11/20/2001 at 12:59:53 From: Doctor Rob Subject: Re: Pythagorean triples Thanks for writing to Ask Dr. Math, Lynn. Both of these equations are much more difficult than the situation with Pythagorean triples. Albert H. Beiler, _Recreations in the Theory of Numbers - The Queen of Mathematics Entertains_ (New York: Dover Publications, 1964), says, on page 220, "The more general relation x^3 + y^3 + z^3 = t^3 also has an infinitude of solutions, the most simple of which are 3^3 + 4^3 + 5^3 = 6^3 and 1^3 + 6^3 + 8^3 = 9^3. There is a fairly complicated formula for all the solutions of this equation, but the formula for all those of a certain type is simple enough to find a place here. It is the identity: a^3(a^3+b^3)^3 = b^3(a^3+b^3)^3+a^3(a^3-b^3)^3+b^3(2a^3-b^3)^3. When a = 2, b = 1, this gives 18^3 = 9^3 + 12^3 + 15^3, which reduces to 6^3 = 3^3 + 4^3 + 5^3. When a = 3, b = 1, we obtain 84^3 = 28^3 + 53^3 + 75^3, and for a = 3, b = 2, we have 105^3 = 33^3 + 70^3 + 92^3. Another identity is a^3(a^3+2b^3)^3 = a^3(a^3-b^3)^3+b^3(a^3-b^3)^3+b^3(2a^3+b^2). Here for a = 2, b = 1, we have 20^3 = 7^3 + 14^3 + 17^3, and for a = 3, b = 1 there results 87^3 = 26^3 + 55^3 + 78^3. Another relation yields 11^3 + 15^3 + 27^3 = 29^3." The equation a^4 + b^4 + c^4 = d^4 was conjectured by Leonhard Euler in the 18th century to have no solutions. No proof was forthcoming, nor were any solutions found until just a few years ago, when Noam Elkies found a fairly large solution, using the theory of elliptic curves. The smallest one was produced by Roger Frye using a computer search: 414560^4 + 217519^4 + 95800^4 = 422481^4. No general formulae are known, nor is it known whether or not the number of solutions is infinite (although that is what I would guess). Feel free to write again if I can help further. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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