# The exponential Diophantine equation $a^n-b^m=x^3+y^3$ for arbitrary large $n,m$

Let $a,b$ be coprime integers, neither a perfect power, $n,m$ naturals and $x,y$ integers.

Consider the exponential Diophantine equation $$a^n-b^m=x^3+y^3 \qquad (*)$$

Nontrivial solution satisfies $|x^3| \ne |a^n|,|x^3| \ne |b^m|, |y^3| \ne |a^n|,|y^3| \ne |b^m|$.

Q1 For fixed $a,b$ are there nontrivial solutions with $n$ and $m$ arbitrary large?

Q2 are there infinitely many pairs $a,b$ for which Q1 gives positive answer?

Numerical evidence suggest these are solutions for all $k,b$: $(a,b,n,m)=(3,b,6,12k)$ and $(3,b,12k+6,24)$.

If this is true, likely the reason is polynomial identity or families of polynomial identities.

In case indeed identities give positive answer:

Q3 Are there more than two univariate identities, preferably infinitely many?

Example:

$3^{66}-5^{24} = 3228113450^3+ 31369668984^3$

We have the following polynomial identity: $$3^{12k+6} - b^{24\ell} \ = \ (3^{3k+2} b^{2\ell} - b^{8\ell})^3 + (3^{4k+2} - 3^{k+1} b^{6\ell})^3.$$ Therefore Question 1 has a positive answer for $a = 3$ and arbitrary $b$, and consequently Question 2 has a positive answer as well. Furthermore, as the above identity is a bivariate identity, it gives rise to infinitely many univariate identities, which also yields a positive answer to Question 3.