Pillai's Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a fixed positive integer $k$, let \[ S_k=\{(m,n)\in\mathbb N^2:|2^m-3^n|=k\}. \] Both exponents are positive, and both signs of the difference are allowed. The bases two and three stay fixed as $k$ varies.

Statement recorded in the supplied source.

For every $k\in\mathbb N$, is $S_k$ finite? The fixed-base assertion is a consequence of the finite-rank multiplicative-group equation theorem cited below: for each sign, $2^m-3^n=\pm k$ becomes a two-variable nondegenerate linear equation with variables $2^m,3^n$. This question does not allow the bases themselves to vary. [S2]

Short English statement

For each fixed positive difference, can only finitely many powers of two and powers of three have that difference in absolute value?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1157 (NT-023). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Jan-Hendrik Evertse, Hans Peter Schlickewei and Wolfgang M. Schmidt, Linear equations in variables which lie in a multiplicative group, Annals of Mathematics 155 (2002), 807--836, Theorem 1.1; arXiv:math/0409604. https://arxiv.org/abs/math/0409604.