(adsbygoogle = window.adsbygoogle || []).push({}); Integers. I need help on this one please???

Hi, i am trying to analyze the following problem, but i am new at abstract algebra so i am not sure whether i am reasoning properly.

Problem:

Why are there no integers x and y with x^2-y^2=34.

Here is how i am reasoning:

x^2-y^2=(x-y)(x+y)=34. so we notice that x-y and x+y are both integral divisors (or factors )of 34.

We know that all factors of 34 are 1,2, 17,34, -1,-2,-17,-34.

This way we have to chose x-y and x+y in such a manner that their product to equal 34. such possibilities are

x-y=1 and x+y=34 , x-y=2 and x+y=17, etc. we take all the cases, i am not typing all of them, but none of these systems of equation has a solution for both x and y integers. so this way we conclude that there are no integers x and y such that x^2-y^2=34.

I want to know whether there is another more elegant way of showing this, rather than what i have performed here?

If you can give some suggestions i would appreciate it. But remember, all we have done so far is 10 pages, we havent even yet gone to common divisors. so don't use too advanced math tools.

Thanx in advance.

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# Integers. I need help on this one please?

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