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Is it known that there exist no coprime positive integers $A$, $B$ and $C$ such that $A^3+B^4=C^3$? This is a particular case of Beal's Conjecture.

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    $\begingroup$ I'm voting to close this question as off-topic because it is an unspecific question around a well-known open problem. $\endgroup$
    – user9072
    Jul 31, 2015 at 13:35
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    $\begingroup$ Well, I disagree. Why is the question unspecific? The OP is asking if a particular case of the conjecture has been settled, and actually it has. $\endgroup$ Jul 31, 2015 at 13:37
  • $\begingroup$ @FrancescoPolizzi as it seems you want to disagree with me it would have seemed like common courtesy to notify me of this fact. $\endgroup$
    – user9072
    Jul 31, 2015 at 13:47
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    $\begingroup$ No it is not different, as the answer below spells out. This very result is mentioned there (it quotes another article but that article mentions the result in the answer for this case). $\endgroup$
    – user9072
    Jul 31, 2015 at 14:12
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    $\begingroup$ @quid: I agree with you that general questions on famous open problems should be closed. Nonetheless, it is my opinion that precise questions on particular cases of famous problems can be accepted. But I guess that this is open to debate. $\endgroup$ Jul 31, 2015 at 14:28

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You can rewrite the diophantine equation in the form $C^3-A^3 = B^4,$ that is $C^3+ (-A)^3=B^4.$

Now, Bruin proved that integer solutions to $x^3+y^3 = z^n$ with $n \in \{4, \, 5\}$ and $xyz \neq 0$ satisfy $\textrm{gcd}(x, \, y, \, z) >1$, see

N. Bruin, On powers as sums of two cubes, Lecture Notes in Computer Science Volume 1838 (2000), pp 169-184.

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  • $\begingroup$ Thanks. I guess I could now ask the question for exponents (3,4,5), which seems to be the new leading unproven candidate, but don't want to give @quid an ulcer! $\endgroup$
    – bobuhito
    Jul 31, 2015 at 15:12
  • $\begingroup$ @bobuhito don't worry about me. Please go ahead. I will just ignore it. $\endgroup$
    – user9072
    Jul 31, 2015 at 15:21
  • $\begingroup$ @bobuhito: quid is right saying that the cases in wich Bael's Conjecture has been proven are easily find with a Google search. For instance, Beuker's paper cited in the Wikipedia's article and the references given therein are a good starting point. If you do not find anything about a specific set of exponents, it is likely that that case is currently unsettled. $\endgroup$ Jul 31, 2015 at 15:52
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    $\begingroup$ (3,4,5) was settled by Samir Siksek and myself: "Partial descent on hyperelliptic curves and the generalized Fermat equation $x^3 + y^4 + z^5 = 0$", Bull. London Math. Soc. 44, 151-166 (2012). It's also on arXiv: arxiv.org/abs/1103.1979 . $\endgroup$ Aug 2, 2015 at 19:54

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