theorem on sums of two squares by Fermat
Suppose that an odd prime number can be written as the sum
Since , the congruence
has a solution , whence
Consequently, the Legendre symbol is , i.e.
Therefore, we must have
where is a positive integer.
The theorem implies easily the
Corollary. If all odd prime factors of a positive integer are congruent to 1 modulo 4 then the integer is a sum of two squares. (Cf. the proof of the parent article and the article “prime factors of Pythagorean hypotenuses (http://planetmath.org/primefactorsofpythagoreanhypotenuses)”.)
|Title||theorem on sums of two squares by Fermat|
|Date of creation||2014-10-25 17:44:02|
|Last modified on||2014-10-25 17:44:02|
|Last modified by||pahio (2872)|