There do not exist positive integers $m$ and $n$ such that $5m + 3n^2 = 15.$

Question

There do not exist positive integers $m$ and $n$ such that $$5m + 3n^2 = 15. $$

Are you trying to type $5m+3n^2=15$? If $m,n>0$ then there really aren't very many things to check. — lulu, Jan 23 '20 at 20:15
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. — Shaun, Jan 23 '20 at 20:22

score 1 · Answer 1 · answered Jan 23 '20 at 20:40

Well, if the integers must be POSITIVE then if $m \ge 3$ then $5m \ge 15$ and $5m + 3n^2 > 15$.

So $m = 1$ or $2$.

And if $n \ge 3$ then $5m + 3n^2 > 27 >15$ so $n=1$ or $2$.

So there are only four cases and none of them work.

Are you sure that the weren't supposed to be no integers at all (which isn't true as $n=0$ and $m=3$ is a solution.

Are maybe the question is suppose to be that there is no integers $m,n$ so that $5m^2 + 3n^2 = 15$.

As $3$ and $5$ divide $15$ and $5$ divides $5m^2$ we'd have $5$ must divide $3n^2$. Which means.....?

And as $3$ divides $5m^2$ which means.....?

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