I'm interested in equations of the form $n = x^a - y^b$, with integers $x, a, y, b, n$, and $a,b > 1$, for different values of $n$. For example:
$1 = 3^2 - 2^3$
$2 = 3^3 - 5^2$
$3 = 2^7 - 5^3$
$4 = 2^3 - 2^2$
$5 = 3^2 - 2^2$
So far, my rather primitive Python program has not found any such equation for $n = 6$. It seems likely that for any given integer, there are two powers of integers out there that differ by that integer. Has this been proven to be true or false?
On
Pillai's Diophantine equation is given by $$ n=a^x-b^y $$ for integers $a,b,x,y,n$. This is a classical topic with many results and several conjectures. A. Herschfeld showed that if $n$ is an integer with sufficiently large $|n|$, then the equation $$ 2^x − 3^y = n $$ has at most one solution $(x, y)$ in positive integers $x$ and $y$. For $|n|\le 10$ he showed that only for $n=-1,1,-5,5,-7,7$ there is a solution.
I think this is probably false in general, but just because OEIS states 6 has no such solutions. It's unclear to me if that's actually been proven or just that no solutions have been found after a large search, but I suspect the former. In any case, you should search for material related to Pillai's conjecture.