# Diophantine equation

Show that the Diophantine equation x^2-y^2=n is solvable in integers if and only if n is odd or n is divisible by 4. When this equation is solvable, find all integer solutions.

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Show that the Diophantine equation x^2-y^2=n is solvable in integers if and only if n is odd or n is divisible by 4. When this equation is solvable, find all integer solutions.

First, we will prove the statement:

x^2-y^2=n is solvable in integers if and only if n is odd or n is divisible by 4.

We need to show that "if x^2-y^2=n is solvable in integers, then n is odd or n is divisible by 4"

Proof. We know that and have the same parity since (x+y)+(x-y)=2x is even. Since x^2-y^2=n is solvable in integers and , we have

Case 1: x+y is odd

Then x-y must be odd. So, is odd.

Case 2: x+y is even

Assume that . Then x-y must be even, say . So,

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#### Solution Summary

This show show to prove that a Diophantine equation is solvable in integers if and only if certain circumstances are met.