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#### sweatingbear

##### Member

- May 3, 2013

- 91

**Problem:**Prove that there exist no positive integers \(\displaystyle x\) and \(\displaystyle y\) such that \(\displaystyle x^2 -y^2 = 270\).

**Proof:**Given that \(\displaystyle x, y \in \mathbb{N}\) where \(\displaystyle \mathbb{N} = \{ 1, 2, 3, \ldots \}\), we can infer that \(\displaystyle x\) and \(\displaystyle y\) respectively will be either even or odd. That means we have four different cases to examine.

*Case #1: \(\displaystyle x\) even, \(\displaystyle y\) even.*Let \(\displaystyle x = 2k \) and \(\displaystyle y = 2p\) (\(\displaystyle k,p\in\mathbb{N}\)); this yields

\(\displaystyle (2k)^2 - (2p)^2 = 270 \iff 4k^2 - 4p^2 = 270 \iff k^2 - p^2 = \frac {270}4 \, .\)

Note now that \(\displaystyle k^2 - p^2\) itself is in an integer but \(\displaystyle \frac {270}4\) is not, which clearly is nonsensical. \(\displaystyle k\) and \(\displaystyle p\) can therefore not be integers and consequently neither \(\displaystyle x\) and \(\displaystyle y\) (at least not even integers).

*Case #2: \(\displaystyle x\) even, \(\displaystyle y\) odd.*Let \(\displaystyle x = 2k \) and \(\displaystyle y = 2p-1\) (\(\displaystyle k,p\in\mathbb{N}\)); this yields

\(\displaystyle (2k)^2 - (2p-1)^2 = 270 \iff 4(k^2 - p^2 + p) - 1 = 270 \iff k^2 - p^2 + p = \frac {271}4 \, .\)

Similarly, \(\displaystyle k^2 - p^2 + p\) is integral but \(\displaystyle \frac {271}4\) is not; nonsensical of course and thus \(\displaystyle k\) and \(\displaystyle p\) cannot be integral (and consequently neither \(\displaystyle x\) and \(\displaystyle y\)).

Note: the case \(\displaystyle x\) odd and \(\displaystyle y\) even is analogous and will therefore be skipped.

*Case #3: \(\displaystyle x\) odd, \(\displaystyle y\) odd.*Let \(\displaystyle x = 2k - 1 \) and \(\displaystyle y = 2p-1\) (\(\displaystyle k,p\in\mathbb{N}\)); this yields

\(\displaystyle 4(k^2 - k - p^2 + p) = 270 \iff k^2 - k - p^2 + p = \frac {270}4 \, .\)

Again, this leads us to the conclusion that \(\displaystyle k\) and \(\displaystyle p\) cannot be integers and therefore neither \(\displaystyle x\) and \(\displaystyle y\). \(\displaystyle \text{Q.E.D.}\)

Any thoughts?