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Is this proof correct:

An odd integer $n \in \mathbb{N}$ is composite iff it can be written in the form

$n = x^2 - y^2, y+1 < x$

Proof:

$\leftarrow$

Want: $n = ab$ Where $a$ and $b$ are odd integers (since $n$ is odd)

Let $n = x^2 - y^2, x > y + 1$. Let $x = \dfrac{a+b}{2}$ and let $y= \dfrac{a-b}{2}$ where $a$ and $b$ are odd integers.

Consider $n = x^2 - y^2$:

$ = (x+y)(x-y) \iff (\dfrac{a+b}{2} + \dfrac{a-b}{2})\cdot(\dfrac{a+b}{2} - \dfrac{a-b}{2})$

Thus we have $ab$.

Now I could do similar steps backwards to prove the other direction.

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    $a \neq 1$ and $b \neq 1$2012-09-30

1 Answers 1

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The part when you say "Let $x=\frac{a+b}2$ and let $y$..." is not really clear. In the $\Leftarrow$ direction $x$ and $y$ should be considered as given, and define $a$ and $b$ using them, and show that they are integers.