Equivalent Sums of SquaresDate: 07/20/2002 at 21:11:40 From: Richard Davies Subject: Sums of Squares Dear Dr. Math, Is it possible for a^2 + b^2 = c^2 + d^2 where a, b, c, d are positive real integers and where the pairs of squares are not identical? If so, are there cases where the result of the sum of these two pairs of squares is itself a square of a real integer? If this were so then it would mean that one could have two right angled triangles with the same integer length hypotonuse but with differing pairs of other integer length sides. I was just curious to know if any one has looked at this. Best regards, Richard Davies. Date: 07/20/2002 at 22:35:44 From: Doctor Paul Subject: Re: Sums of Squares It is indeed possible, and much has been written on the subject. The smallest integer that can be written as the sum of two squares in two different ways is: 50 = 1^2 + 7^2 = 5^2 + 5^2 Other numbers on the list include: 65, 85, 125, 130, 145, 170, 185, 200, 205, 221, 250, 260, 265, 290, 305, 338, 340, 365, 370, 377, 410, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 625, 629, 680, 685, 689, 697, 730, 740, 745, 754, 765, 785, 793, 800, 820, ... Notice that 625 = 25^2 is on the list. The list doesn't say how to write the number on it as the sum of two squares in two different ways, but a computer is quick to find the answer: ? for(n=1,24,print(n," ",sqrt(625-n^2))) 1 24.97999199359359282338757248 2 24.91987158875422455775278823 3 24.81934729198171319226648391 4 24.67792535850613192708368654 5 24.49489742783178098197284074 6 24.26932219902319398106317633 7 24 8 23.68543856465402246658547991 9 23.32380757938120188349661150 10 22.91287847477920003294023596 11 22.44994432064364831350249239 12 21.93171219946130881654807169 13 21.35415650406262242162304793 14 20.71231517720797913215717634 15 20 16 19.20937271229854605946465302 17 18.33030277982336002635218877 18 17.34935157289747241232499427 19 16.24807680927192072091976713 20 15 21 13.56465996625053627812911265 22 11.87434208703791723467291760 23 9.797958971132712392789136298 24 7 Thus: 625 = 25^2 = 7^2 + 24^2 = 15^2 + 20^2 I think you'll find more than enough information to occupy a few hours of your time in Eric Weisstein's MathWorld: Square Number http://mathworld.wolfram.com/SquareNumber.html Take a look at the table about 60% of the way down the page. The case W = S = 2 is what you're asking about. The other cases may be interesting to you also. I hope this helps. Please write back if you'd like to talk about this some more. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/ |
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