hard diophantine equation: $x^3 + y^5 = z^7$ Does the equation $x^3+y^5=z^7$ have a solution $(x,y,z)$ with $x,y,z$ positive integers and $(x,y)=1$? In his book H. Cohen (Number theory,2007) said "[...] seems presently out of reach".
I couldn't find any suggestion beyond Cohen's book.
Thanks in advance,
Montanari Fabio
department of math
university of bologna
italy
e-mail montana@dm.unibo.it
 A: Sander Dahmen and Samir Siksek are [Edit] writing a paper [end Edit] about this (according to Samir's cv, under papers in preparation), but there is no draft on his web page.
A: I'm surprised that Bjorn Poonen hasn't chimed in yet.  However, you can read about the approaches to solving equations like this in Frits Beuker's lectures "The generalized Fermat Equation" http://www.math.uu.nl/people/beukers/Fermatlectures.pdf
Added later: You should also look at Siksek's "Edingurgh lectures" for a good outline of how he might have approached this equation: http://www.warwick.ac.uk/~maseap/papers/edinburgh3.pdf
A: I may be misunderstanding the question, but I do not believe that it has any integer solutions.  At the very least, none are known to exist at the moment.  Any solutions would be counterexamples to the Fermat-Catalan conjecture with {m,n,k} = {3,5,7} (since 1/3 + 1/5 + 1/7 = 71/105 < 1).  The most I can tell you is that, for coprime {x,y,z}, there are finitely many solutions to your equation.  I think your (x,y) = 1 means that they're coprime, anyway, so it follows that z must be coprime.  Therefore, any solution at all would disprove the related Beal's conjecture.
A: Hi,
There is no claim in my cv or elsewhere that me and Sander have solved the equation x^3+y^5+z^7=0. All my cv claims is that we're writing a paper on it! That's not the same thing.
All the best,
Samir
