Pairwise Coprime Nonrepresentable Integers for Two Coprime Bases

Abstract

Let p > q ≥ 2 be coprime integers. Call a positive integer (p, q)-nonrepresentable if it cannot be written as a sum of terms p^i q^j for which no two distinct terms divide each other. Erdős and Lewin asked for which pairs there is an infinite sequence of pairwise coprime (p, q)-nonrepresentable integers. Yu and Chen answered this question except for the pairs (5, 2), (9, 2), and (5, 3). We prove the remaining cases. The proof works by creating a criterion for generating such sequences, which can then be applied to the three pairs.

Citation

Tadayoshi Kamegai. Pairwise Coprime Nonrepresentable Integers for Two Coprime Bases. Preprint, 8 October 2026.

https://apiros3.github.io/Notes/publication/pairwise-coprime-nonrepresentable-integers.html

@misc{pairwise-coprime-nonrepresentable-integers,
  author = {Tadayoshi Kamegai},
  title = {Pairwise Coprime Nonrepresentable Integers for Two Coprime Bases},
  year = {2026},
  note = {Preprint, 8 October 2026},
  url = {https://apiros3.github.io/Notes/publication/pairwise-coprime-nonrepresentable-integers.html}
}