Effective determination of solutions to a mixed-power Diophantine equation

Determine all finitely many non-negative integer solutions to the Diophantine equation $3^a+5^b-7^c=1$.

Background

The paper explains that deciding emptiness of intersections of pp-normal sets associated with three or more distinct primes reduces to a difficult existential first-order theory involving multiple power predicates. This general decision problem already encompasses unresolved Diophantine questions, including the determination of all non-negative integer solutions of the displayed equation.

References

The case of $k \geq 3$ already subsumes longstanding open problems in number theory, such as finding all the finitely many non-negative integer solutions to the Diophantine equation $3a + 5b - 7c = 1$.

— Skolem-Mahler-Lech in rings of positive characteristic: a shorter proof and a multi-dimensional generalization  (2609.03127 - Dong et al., 2 Sep 2026) in Section 4, subsection “$p$-normal sets in $N^n$”