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I have to prove that the equation $$15x^3+13y^3=101$$ does not have any solutions over the integers. Can you provide any hint?

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    What did you try? Maybe this is a hard diophantine equation or ... maybe it's an easy one. In the later case, probably you can solve it using modular arithmetic. – 2017-01-12
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    Mod $9$ does not work, but mod $7$ does. – 2017-01-12
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    For the OP: many users here tend to downvote so-called PSQs (Problem Statement Questions) like yours, and answers, too (some god will forgive them, but the poor human Jack won't). So it is in your (and community's) best interest to improve your actual question by adding some context (your attempts, why this question is relevant to you, something along these lines). Cheers. – 2017-01-12

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$\pmod{7}$, a perfect cube $\in\{0,1,6\}$. That implies: $$ 15x^3+13y^3 \equiv x^3+(-y)^3 \in \{0,1,2,5,6\} \pmod{7}$$ while $101\equiv\color{red}{3}\pmod{7}$.

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    Give a man a fish (and he will continue begging for more)... – 2017-01-12
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    x^6 is 0 or 1 mod 7 by fermat's little theorem, so x^3 is 0 or 1 or -1 mod 7 because x^6 - 1 = (x^3+1)(x^3-1) and euclid's lemma. – 2017-01-12
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    @barakmanos: you are probably right, if I correctly understand your irony. Anyway, I believe the process of learning math actually involves a pretty large amount of *begging* (for ideas, for complete solutions, whatever), and I think helping people cannot be too wrong, assuming the involved people are in good faith :) – 2017-01-12
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    I agree, but I think that you just couldn't resist the temptation of solving yet another problem (I know, because I sometimes find myself unable to resist the exact same temptation myself). But you did it in the expense of allowing this OP to get away with a not-even-polite request for help (more precisely, a command for help). Now, ask yourself - if that was your spoiled kid asking something in a similar manner - would you let them "get away with it" and grow up to be a spoiled teenager (and ultimately a spoiled adult)? – 2017-01-12
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    @barakmanos: well, manners can be taught, too. I accept the suggestion, let we try to do that. – 2017-01-12
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    Are you kidding me? You've changed OP question to project a more polite manner??? Let me bring you some shocking news - that ain't gonna teach this user any manners. – 2017-01-12
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    How do you teach kids how to behave? By telling them, and showing them, how to behave. Can it really work in any different way? – 2017-01-12
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    Yes, but I don't see you "planting" words in their mouth (that's not even possible). In any case, this is an educative debate which is off-context for this website I suppose, so no point continuing it. Mathematically, your answer is great, so I'll leave it off with a +1... – 2017-01-12