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The Diophantine equation $$x^2+y^3=z^2$$ has solutions $(\pm 1,2,\pm 3)$ and $(\pm 13,3,\pm 14)$.

Brown [Int. Math. Res. Not. IMRN 2012, no. 2, 423–436; MR2876388] states that "there are infinitely many parameterized solutions with $xyz \neq 0$ and $z^2 \neq 1$ when $n \leq 5$" to the equation $$x^2 +y^3 = z^n,$$ but does not give a reference. I am looking for a reference that catalogs all primitive solutions.

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    $\begingroup$ You can treat this as a special case of factoring, e.g. $Y=(z+x)(z-x)$, where you can have one of the factors (say $z-x$) be $y$ to a power. If you want everything integral, make sure the factors have the same parity. Gerhard "Difference Of Squares Is Easy" Paseman, 2017.03.14. $\endgroup$
    – Gerhard Paseman
    Commented Mar 15, 2017 at 2:33
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    $\begingroup$ The last, and by far the hardest, case is $n=5$, which is solved in: J. Edwards: A Complete Solution to $X^2 + Y^3 + Z^5 = 0$, Journal f. d. reine und angew. Math. (Crelle's Journal) 571 (2004), 213-236. (Change $(X,Y,Z)$ to $(x,y,-z)$ for your form of the equation.) That paper should also give references for $n<5$. $\endgroup$
    – Noam D. Elkies
    Commented Mar 15, 2017 at 3:21
  • $\begingroup$ @NoamD.Elkies: there are references but they concern the cases when $3 \leq n \leq 5$. $\endgroup$
    – Pietro Paparella
    Commented Mar 15, 2017 at 3:57
  • $\begingroup$ The parameterization equation $x^2+y^3=z^2$ can record such. artofproblemsolving.com/community/c3046h1217955_the_cube_view For $x^2+y^3=z^4$ artofproblemsolving.com/community/… As already said - it is better for each individual degree to solve directly the equation. $\endgroup$
    – individ
    Commented Mar 15, 2017 at 4:44

2 Answers 2

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All solutions of the Diophantine equation $x^2-y^2=z^3$ can be found in the book "Number Theory. Volume II: Analytic and Modern Tools" (Springer Science, 2007) by Henri Cohen. Below the relevant page from this book is reproduced.

As for general case of the Generalized Fermat equation, see https://www.staff.science.uu.nl/~beuke106/Fermatlectures.pdf The generalized Fermat equation, by Frits Beukers), http://homepages.warwick.ac.uk/~maseap/papers/bealconj.pdf (The Generalized Fermat Equation, by Michael Bennett, Preda Mihailescu and Samir Siksek) and http://people.math.sfu.ca/~ichen/pub/BeChDaYa.pdf (Generalized Fermat equation: a miscellany, by M.A. Bennett et al.).

page 465 from the Cohen's Number Theory book (Vol. 2)

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A warm-up:

$$ \left(\frac{s^n-t^n}2\right)^2 +\ (s\cdot t)^n\ =\ \left(\frac{s^n+t^n}2\right)^2 $$

provides integer solutions for $\,\ s\equiv t \mod 2 $.

(Sorry, I couldn't help it).

 

REMARK   More generally, going in the abc direction:

$$ \left(\frac{s^m-t^n}2\right)^2 +\ s^m\cdot t^n\ =\ \left(\frac{s^m+t^n}2\right)^2 $$

for $\ s\equiv t\equiv 1 \mod 2,\ $ and $\ \gcd(s\ t) = 1$.

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    $\begingroup$ In the remark, should it be $s \equiv t \equiv 1 \bmod 2$? $\endgroup$
    – Pietro Paparella
    Commented Mar 15, 2017 at 16:46
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    $\begingroup$ @PietroPaparella, yes, mod 2--thank you, I'll fix it. $\endgroup$
    – Włodzimierz Holsztyński
    Commented Mar 15, 2017 at 17:11

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