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The problem asks to prove that the Diophantine equation $x^{3}+y^{3} = (x+y)^{2}+(xy)^{2}$ does not have any solutions in natural numbers $x, y$.

I believe that this problem appeared in the section Задачи наших читателей of the Soviet magazine Квант somewhere between the first issue of 1980 and the last one of 1989. Since I don't know much Russian, I haven't been able to locate it by surfing the archives of the magazine that are available online: to add insult to injury, it seems to me that the section in question of the magazine was not a regular one. I would like to provide the exact reference for this problem in a certain document which I am preparing and that's the main reason that has compelled me to ask you this:

Did anybody here remember seeing this cute problem in Квант once? If so, would you be so kind as to provide me with a hint that allows one to find out what the actual issue wherein it appeared was?

Please, let me thank you in advance for your attentive consideration of this query of mine.

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    $\begingroup$ Where is variable $z$? $\endgroup$ – individ Nov 28 '16 at 8:11
  • $\begingroup$ @individ: I have just corrected the statement of the problem... Thanks a lot for pointing this out! $\endgroup$ – José Hdz. Stgo. Nov 28 '16 at 8:14
  • $\begingroup$ So you are not sure if this problem existed in this journal at all? Why don't you just solve the equation?(it does not seem too difficult) $\endgroup$ – Konstantinos Gaitanas Nov 28 '16 at 10:33
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    $\begingroup$ @KonstantinosGaitanas: It is not that I cannot solve it, it's just that I am really interested in determining the provenance of it. $\endgroup$ – José Hdz. Stgo. Nov 28 '16 at 11:03
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It appeared in issue 8 of 1984 at the page 34.You can download this issue from here: http://kvant.mccme.ru/oblozhka_djvu3.htm

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    $\begingroup$ The author of this problem is M.Garaev matmor.unam.mx/~garaev $\endgroup$ – Alexey Ustinov Nov 28 '16 at 12:18
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    $\begingroup$ By the way, does anybody now how to solve problem 2 on the same page, by A. T. Kurgansky, that is, to prove that for positive integers $a,b$ and a prime number $p>max(a,b)$ the number $p^3$ never divides $(a+b)^p-a^p-b^p$ ? I have no idea already for $a=b=1$. $\endgroup$ – Fedor Petrov Nov 28 '16 at 13:04
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    $\begingroup$ @FedorPetrov : A Theorem of Vandiver seems relevant to your question, but it does not seem to prove that $2^{p-1} \equiv 1$ ( mod $p^{3}$) is impossible, it just gives a necessary and sufficient condition for it to hold. The theorem in the Wikipedia article on Wieferitz primes, for example, mentions Vandiver's Theorem, but it was unclear to me what the exact statement was. $\endgroup$ – Geoff Robinson Nov 28 '16 at 17:18
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    $\begingroup$ A typo: WieferiCH primes en.wikipedia.org/wiki/Wieferich_prime $\endgroup$ – Alexey Ustinov Nov 29 '16 at 4:07
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    $\begingroup$ Vandiver's Theorem is here, see page 112 jstor.org/stable/pdf/2007115.pdf $\endgroup$ – Alexey Ustinov Nov 29 '16 at 4:19

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