Does $a^6+b^6 = c^6+d^3$ have a non-trivial solution?

Question

It is conjectured that,

$$x_1^8+x_2^8+x_3^8 = y_1^8+y_2^8+y_3^8\tag{1}$$

has no non-trivial solutions. However, if we relax it a bit, then,

$$x_1^8+x_2^8+x_3^4 = y_1^8+y_2^8+y_3^4\tag{2}$$

can be shown to have an infinite number of primitive solutions such as,

$$86^8+149^8+14805^4=35^8+142^8+18939^4$$

Likewise, it is conjectured that,

$$x_1^6+x_2^6 = y_1^6+y_2^6\tag{3}$$

is only trivially solvable. However,

$$x_1^6+x_2^3 = y_1^6+y_2^3\tag{4}$$

also has an infinite number of solutions, such as,

$$5^6+167^3 = 8^6+164^3$$

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Question: Can we relax $(3)$ even further? Is

$$a^6+b^6 = c^6+d^3\tag{5}$$

non-trivially solvable?

Answer 1

Mea culpa. I should have done a Mathematica search before posting this question. Normally I would, but I thought if there was a non-trivial solution to,

$$a^6+b^6 = c^6+d^3\tag{1}$$

then it might be large. Turns out it is small,

$$15^6+18^6 = 19^6 + (-118)^3$$

and is the only primitive one with $a,b,c,< 100$. (Beyond that, it takes too long for my rather slow computer.)

I'll have to revise my question to: Is there a solution $a,b,c,d$ all positive?

Edit (a day later)

I just realized, regardless of the positivity of $a,b,c,d$, that there is an infinite number of primitive solutions to $(1)$ using the well-known identity,

$$(3t^2)^6+(3t-9t^4)^3 = 1+(9t^3-1)^3$$

All one has to do is find rational t such that the second term is also a square,

$$3t-9t^4 = y^2$$

which, after transforming to Weierstrass form, is an elliptic curve. One such point is $t=3/13$ which yields,

$$27^6+138^6 = 13^{12}+(-25402)^3$$

Another is $t = 208098704151/634343459737$ (though presumably there may be points of smaller height), and so on.