If an Even number could be expressed in the form a² + b² . And if there exits two other numbers m,n such that

a² + b² = m² + n²

then , my question is

is there any relation between (a,b) and (m,n) apart from a² + b² = m² + n² ??

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- Apr 6th 2011, 11:59 PMssnmanikantasum of two squares?
If an Even number could be expressed in the form a² + b² . And if there exits two other numbers m,n such that

a² + b² = m² + n²

then , my question is

is there any relation between (a,b) and (m,n) apart from a² + b² = m² + n² ?? - Apr 7th 2011, 05:39 AMOpalg
The condition for an integer x to be a sum of two squares in more than one way is that x should have at least two prime factors of the form 4k+1. For example, , and , with both 5 and 13 being a multiple of 4 plus 1. But there is no particular relation between the pairs of numbers (1,8) and (4,7) occurring in those decompositions.

- Apr 7th 2011, 07:23 AMSoroban
Hello, ssnmanikanta!

Quote:

There is an interesting theorem that may or may not help . . .

If a number is of the form , the product of two sums of squares,

. . then can be expressed as the sum of two squares in two ways.

. .