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I've been told that there is reason to think that the equation

$x^3 + y^3 = z^3 + 3$ has solutions in positive integers other than

$$4^3 + 4^3 = 5^3 +3.$$

Can someone tell me the current status of this problem? Is there another solution known?

Here's a reference to the problem in greater generality.

MATHEMATICS OF COMPUTATION Volume 66, Number 218, April 1997, Pages 841–851 S 0025-5718(97)00830-2 ON SEARCHING FOR SOLUTIONS OF THE DIOPHANTINE EQUATION $x^3 + y^3 + z^3 = n$

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    $\begingroup$ In arxiv.org/abs/1903.04284 it is shown that there are no other with $|x|,|y|,|z|<10^{16}$. $\endgroup$
    – Xarles
    Commented Aug 8, 2019 at 21:59
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    $\begingroup$ Could mention that mathoverflow.net/questions/58188/… has an extensive discussion, as of 2011, of a possibly easier yet still seemingly infeasible problem. $\endgroup$
    – Colin McLarty
    Commented Aug 8, 2019 at 22:01
  • $\begingroup$ Another solution to $x^3 + y^3 + z^3 = 3$ in integers is now known (Sept.2019): $569936821221962380720^3 + (-569936821113563493509)^3 + (-472715493453327032)^3 = 3$. This does not rearrange to $a^3 + b^3 = c^3 + 3$ in positive integers $a, b, c$. $\endgroup$
    – KConrad
    Commented Sep 19, 2019 at 13:40

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