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While studying the properties of a certain elliptic curve, I came across the equation $x^4+y^4=z^3$. There is no solution of this equation in relatively prime integers, and this is a special case of Beal's conjecture.

According to this source, the proof that $x^4+y^4=z^3$ has no solutions in relatively prime integers was given by Lucas in the 19th century. Does anyone have a reference which links to Lucas's proof (or even a document which gives a proof sketch)?

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This paper by Darmon and Merel, which proves the general case $x^n + y^n = z^3$ has no primitive solutions, cites Dickson, "History of the theory of numbers", page 630. For a good resource on the state of the art on the generalized Fermat problem (well, as of 2005), see Poonen, Schaefer, Stoll, "Twists of $X(7)$ and primitive solutions to $x^2+y^3=z^7$". This is where I found the Darmon-Merel reference. If someone knows a good reference with a nice summary of what's known as of a more recent date, please let me know.

As a side note, my impression is that there are many professional mathematicians do not prefer the name Beal's conjecture for this; I have occasionally seen a parallel problem called the "Generalized Fermat conjecture": that $x^n + y^m = z^\ell$ has finitely many primtive solutions $(x,y,z,n,m,\ell)$, where $1/n + 1/m + 1/\ell < 1$ (other than the ones that obviously work for an infinite family of $(n,m,\ell)$; e.g. $(2,\pm 3, 1, 3,2,\ell)$. You will not find the name Beal's conjecture in reputable published papers on the problem, for instance.

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  • $\begingroup$ Thanks for the reference! I'm downloading Dickson's book; when it's done I'll check it out. (Also: the reason I used the name "Beal's conjecture" was because Noam Elkies used it in our emails. But, of course, that's in an informal context!) $\endgroup$ – user122283 Sep 2 '15 at 16:03
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    $\begingroup$ Here is a more modern reference that includes tables of what $(n,m,\ell)$ the problem is solved for. $\endgroup$ – user98602 Sep 2 '15 at 16:08
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The following answer to my question was possible thanks to the links which Mike Miller gave. On page 630 of Dickson's "History of the Theory of Numbers", the following is written: "E. Lucas listed and treated the solvable equations $ax^4+by^4=cz^2$" in which $2$ and $3$ are the only primes dividing $a,b$, or $c$, viz." (he then lists out a bunch of values for $(a,b,c)$). In the fifth section of the second link (this), the relation between Lucas's work and the solutions to the equation $x^4+y^4=z^3$ is given. This is possible because of a certain change of variables which reduces to studying equations of the form $ax^4+by^4=cz^2$, and Lucas's theorem can be applied here.

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