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The generalized Fermat equation has been solved for many signatures. But, I can't find a determination that the signature $[p,q,r]=[4,5,7]$ has no solutions. Is this signature still an open problem?

$\chi=1/4+1/5+1/7<1$

$x^p+y^q=z^r$

$gcd(x,y,z)=1$

$p,q,r\ge3$

The equation has only finitely many solutions with a bound given by the Darmon-Granville Theorem. Many other signatures have already been eliminated like $$ [p,p,p], [3,5,5], [4,5,5], [3,4,7], $$ and others.

Laisham and Shorey's 2011 result in their theorem 4 is that assuming Baker's conjecture there are no solutions to anything not in $$ \{[3,5,𝑝] : 7≀ 𝑝 ≀ 23,𝑝 \text{ prime}\}βˆͺ\{[3,4,𝑝]:𝑝 \text{ prime}\} . $$ This immediately implies at least a conditional elimination for $[4,5,7]$.

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    $\begingroup$ It has no known solutions, and by a general theorem has only finitely many solutions. I think it's still open, as to whether it has any solutions at all. $\endgroup$ – Gerry Myerson May 26 at 4:20
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    $\begingroup$ The finite solution bound is called the Darmon-Granville Theorem. Many signatures have already been eliminated like (p,p,p), [3,5,5], [4,5,5], [3,4,7], and others. The question is only about the signature [4,5,7]. $\endgroup$ – Pythagorus May 26 at 4:29
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    $\begingroup$ I understand the question. My comment is all I know about 4, 5, 7. It may be all that is known about 4, 5, 7. $\endgroup$ – Gerry Myerson May 26 at 4:31
  • $\begingroup$ Laisham and Shorey's 2011 result in their theorem 4 is that assuming Baker's conjecture there are no solutions to anything not in $Q\epsilon\{[3,5,p]:7\le$ $p\le23, p$ prime$\}\cup\{[3,4,p]:p$ prime$\}$ This immediately implies at least a conditional elimination for [4,5,7]. $\endgroup$ – Pythagorus May 26 at 4:55
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    $\begingroup$ It's probably best that when you post a question, you include right away everything you already know about it. If you don't get a satisfactory answer here after a few days, you might consider posting the question to MathOverflow (being certain to include a link at each site to the question at the other site). $\endgroup$ – Gerry Myerson May 26 at 8:48

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