When Alpha is submitted the equation $a(a^2-1)=2b^2$, it unexpectedly forgets the integer solution $a=1,b=0$. What could explain this ?



closed as off-topic by Mariusz Iwaniuk, Michael E2, Szabolcs, m_goldberg, b3m2a1 Nov 10 '18 at 17:34

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    $\begingroup$ Workaround, putt to Wolfram Alfa: solve a*(a^2-1)=2*b^2, a>=0,b>=0, for integers, see last solution? $\endgroup$ – Mariusz Iwaniuk Nov 9 '18 at 16:15
  • $\begingroup$ @MariuszIwaniuk: thanks for the suggestion, but my question is more about why it is so. $\endgroup$ – Yves Daoust Nov 9 '18 at 16:37
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    $\begingroup$ I don't think it makes sense to ever trust a computer algebra system. It is a tool, and it can sometimes do amazing things, but it is certainly not without its quirks. Check and double check everything! $\endgroup$ – bill s Nov 9 '18 at 20:50
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    $\begingroup$ @bills: after several years of usage, this is the first time I see a result that cannot be explained (I have seen several Mathematics posts claiming Alpha mistakes, but in all cases Alpha was right and the output was misinterpreted by the posters). IMO, Mathematica is more reliable than most humans. $\endgroup$ – Yves Daoust Nov 9 '18 at 22:47
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    $\begingroup$ @MichaelE2: then my question is off-topic, as I don't have Mathematica. $\endgroup$ – Yves Daoust Nov 10 '18 at 9:34

I have observed that the anomaly disappears when the resolution is explicitly asked over the integers (the qualifier Diophantine also works).


So it seems that the flaw would be in the logics for the "Integer solutions" section that comes along with an unspecified domain (presumably $\mathbb C$ by default). As suggested by @chiphurst, this could be because a general solver is used and might fail to find the exact integers.


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