Sum of Two Squares: Is There a Relation?

[smslca]

sum 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 ??

 

[HallsofIvy]

Well, there is the obvious $a^2- m^2= n^2- b^2$ so that $(a- m)(a+ m)= (n- b)(n+ b)$.

 

[configure]

Well, there is the obvious $a^2- m^2= n^2- b^2$ so that $(a- m)(a+ m)= (n- b)(n+ b)$.

Or, there is $a^2- n^2= m^2- b^2$.

 

[Char. Limit]

Well, there is the obvious $a^2- m^2= n^2- b^2$ so that $(a- m)(a+ m)= (n- b)(n+ b)$.

Or, there is $a^2- n^2= m^2- b^2$.

So other relations between (a,b) and (m,n) include

$$(a-m)(a+m) = (n-b)(n+b)$$

$$(a-n)(a+n) = (m-b)(m+b)$$

All of the numbers must solve both of these equations.

 

[2003 UB313]

A prime of form 1 (mod 4) can be uniquely written as sum of two squares.

An even number can be uniquely expressed as a sum of two squares if it is twice a prime of form 1 (mod 4). (or if it is multiplied with any square of number of form 3 (mod 4)).

A number which is product of N distinct primes of form 1 (mod 4) can be represented as sum of two squares in N different ways.
Thus, an even number can be written as sum of two squares in N different ways if it is twice that value (or product of that with any square of number that is equivalent to form 3 (mod 4)).

For more information, please see
refer to this thread.
http://mersenneforum.org/showthread.php?t=13027