Show that if p3 mod 4 is any integer then the equation x2y2p

Show that if p=3 mod 4 is any integer, then the equation x^2+y^2=p has no integer solutions.

Let\'s look at residues of squares modulo 4

1^1=1

2^2=0

3^2=1

4^2=0

Hence, x^2=0 or 1 mod 4 ,y^2=0 or 1 mod 4

Hence x^2+y^2 =0 or 1 or 2 mod 4

But ,x^2+y^2=p

Hence,

x^2+y^2=p=3 mod 4

$Show that if p=3 mod 4 is any integer, then the equation x^2+y^2=p has no integer solutions.SolutionLet\'s look at residues of squares modulo 4 1^1=1 2^2=0 3^2=$

Show that if p3 mod 4 is any integer then the equation x2y2p

Solution

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