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# Diophantine equation

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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.

https://brainmass.com/math/algebra/proof-solution-involving-diophantine-equation-26408

#### Solution Preview

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.

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