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PreprintCC BY 4.0arXiv

Positivity and negativity for h-bases for n

  • Wouter van Doorn
  • Pietro Monticone
  • Melvyn B. Nathanson
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Abstract

A set A of integers is an h-basis for n if every integer in [0,n] = {0,1,2,…,n} belongs to the sumset hA, that is, can be represented as the sum of exactly h not necessarily distinct elements of A. The classical postage stamp problem asks for the largest integer nh(k) such that [0,nh(k)] ⊆ hA for some set A of nonnegative integers of size k+1. We consider the effect of allowing negative elements. Let n♭h(k) be the largest integer such that [0,n♭h(k)] ⊆ hA for some set A of integers of size k+1 containing at least one negative integer. For k=2, we have n♭h(2) < nh(2) if h is odd or h=4 and n♭h(2) = nh(2) if h is even and h ≠ 4. Proofs of these comparison results were autonomously discovered and formally verified in Lean by https://aristotle.harmonic.fun/}{Aristotle}, a formal reasoning agent developed by https://www.harmonic.fun/}{Harmonic}. In this paper we give new proofs of these results.

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Cite

BibTeXRIS
BibTeX
@misc{doorn2026positivity,
  title = {{Positivity and negativity for $h$-bases for $n$}},
  author = {Wouter van Doorn and Pietro Monticone and Melvyn B. Nathanson},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19470},
  eprint = {2610.19470},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19470},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Positivity and negativity for h-bases for n
AU  - Wouter van Doorn
AU  - Pietro Monticone
AU  - Melvyn B. Nathanson
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.19470
UR  - https://arxiv.org/abs/2610.19470
AB  - A set A of integers is an h-basis for n if every integer in [0,n] = {0,1,2,…,n} belongs to the sumset hA, that is, can be represented as the sum of exactly h not necessarily distinct elements of A. The classical postage stamp problem asks for the largest integer n_h(k) such that [0,n_h(k)] ⊆ hA for some set A of nonnegative integers of size k+1. We consider the effect of allowing negative elements. Let n^♭_h(k) be the largest integer such that [0,n^♭_h(k)] ⊆ hA for some set A of integers of size k+1 containing at least one negative integer. For k=2, we have n^♭_h(2) < n_h(2) if h is odd or h=4 and n^♭_h(2) = n_h(2) if h is even and h ≠ 4. Proofs of these comparison results were autonomously discovered and formally verified in Lean by https://aristotle.harmonic.fun/}{Aristotle}, a formal reasoning agent developed by https://www.harmonic.fun/}{Harmonic}. In this paper we give new proofs of these results.
ER  -