Diophantine equation of $a^2+b^2=c^2+d^2$

Question

Is there a known solution to:

$$a^2+b^2=c^2+d^2$$

Hopefully the question is clear, if not I do apologize.

Answer 1

There are lots of solutions. One of them is $1^2+8^2=65=4^2+7^2$.

Answer 2

There is a wonderful theorem concerning these kind of numbers:

If $a^2 + b^2 = c^2 +d^2 = E$, then there exist numbers $f,g,h,k$ such that $E= (f^2+g^2)(h^2+k^2)$.

There is a wonderful geometric proof of this. For example: $$ 50= 7^2 + 1^2 = (1^2+1^2)(3^2+4^2) $$

I will not go into the details, but this equation was resoundingly solved by Ramanujan and other mathematicians, by using what are called the q-series, and not the diophantine equation itself. Hence I recommend you to look at these two pages:

1) http://mathworld.wolfram.com/SumofSquaresFunction.html (As a reference)

2)http://www.rowan.edu/colleges/csm/departments/math/facultystaff/osler/110%20SUM%20OF%20TWO%20SQUARES%20IN%20MORE%20THAN%20ONE%20WAY%20MACE%20Small%20changes%20Oct%2008%20%20Submission.pdf (Contains the geometric proof)