Integer as Difference between Two Squares/Formulation 2
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Theorem
Any integer can be expressed as the difference of two squares if and only if that integer is NOT $n \equiv 2 \pmod 4$
Proof
- Each integer will be in one of the 4 sets of residue classes modulo $4$:
- $n \equiv 0 \pmod 4$
- $n \equiv 1 \pmod 4$
- $n \equiv 2 \pmod 4$
- $n \equiv 3 \pmod 4$
\(\ds 4x\) | \(=\) | \(\ds \paren{x + 1 }^2 - \paren{x - 1 }^2\) | ||||||||||||
\(\ds 4x + 1\) | \(=\) | \(\ds \paren{2x + 1 }^2 - 4x^2\) | ||||||||||||
\(\ds 4x + 3\) | \(=\) | \(\ds \paren{2x + 2 }^2 - \paren{2x + 1 }^2\) |
- For $n \equiv 2 \pmod 4$, it is impossible to represent such an integer as the difference of two squares.
Taking the squares of each of the residue classes, we have:
- $0^2 \equiv 0 \pmod 4$
- $1^2 \equiv 1 \pmod 4$
- $2^2 \equiv 0 \pmod 4$
- $3^2 \equiv 1 \pmod 4$
- Therefore, when taking the difference of two squares, $n \equiv 2 \pmod 4$ is never a result.
- $0 - 0 \equiv 0 \pmod 4$
- $1 - 1 \equiv 0 \pmod 4$
- $0 - 1 \equiv 3 \pmod 4$
- $1 - 0 \equiv 1 \pmod 4$
$\blacksquare$