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Do there exist positive integers $a, b, c, d, e, f$ such that $a^{2}b^3 + c^{2}d^3 = e^{2}f^3$ where $b, d, f$ are pairwise coprime ?

Addendum: From the comments and Matt. F's answer, there clearly are infinitely many solutions. But what are their parametrizations ?

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    $\begingroup$ $1^{3}+2^{3} = 3^{2}$ ($a=b = c = f =1$ and $d = 2, e = 3$). $\endgroup$
    – Geoff Robinson
    Commented Aug 24, 2020 at 9:19
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    $\begingroup$ And even infinitely many: take $c=f=x+1,d=e=x, b=1, a=x(x+1) $, for $x=2$ this is Matt F.'s example. $\endgroup$
    – Fedor Petrov
    Commented Aug 24, 2020 at 9:44
  • $\begingroup$ @FedorPetrov, okay, can the solutions be parametrized ? $\endgroup$
    – Q_p
    Commented Aug 24, 2020 at 10:26
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    $\begingroup$ $a^{2}b^3 + c^{2}d^3 = e^{2}f^3\implies \Big(\dfrac{ab^3}{c}\Big)^2 - (bf)^3\Big(\dfrac{e}{c}\Big)^2 = -(bd)^3$ $$$$ This have Pell form. Some solutions $(a,b,c,d,e,f)$=(137819, 7, 7, 5, 491218, 3), (1522899144, 7, 27, 4, 1044610624, 9), (5925421773487638370, 7, 70, 3, 3470294476762229557, 10). $\endgroup$
    – Dmitry Ezhov
    Commented Aug 24, 2020 at 10:30
  • $\begingroup$ $r^3s^2(r+s)^2+r^2s^3(r+s)^2=r^2s^2(r+s)^3$ provided $\gcd(r,s)=1$. $\endgroup$
    – Gerry Myerson
    Commented Aug 24, 2020 at 12:27

2 Answers 2

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Yes: $6^2 1^3+3^2 2^3=2^2 3^3$

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We can get the parametric solutions using known solution $(a,b,c,d,e,f)$ for fixed $(b,d,f).$
For instance, we get a parametric solution using $(a,b,c,d,e,f)=(6,1,3,2,2,3).$
$1^3(48m^2-6n^2-48mn)^2 + 2^3(24m^2-3n^2+12mn)^2 = 3^3(16m^2+2n^2)^2$
$m,n$ are arbitrary.

             m  n     a     c     e
             
             1  1     2    11     6
             1  2     6     3     2
             1  3   150    33    34
             1  4    10     1     2
             1  5   114     3    22
             2  1    30    39    22
             2  3   150   141    82
             2  5   146    47    38
             3  1   282   249   146
             3  2    30    69    38
             3  5   438   321   194
             4  1   190   143    86
             4  3   138   501   274
             4  5    38    61    34
             5  1   318   219   134
             5  2    58    59    34
             5  3   426   753   418
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