Equivalent Sums of Squares

Date: 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/