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PreprintCC BY-NC-ND 4.0arXiv

Periodicity conjectures for all 2-sumfree sequences

  • Daan van Berkel
  • Wieb Bosma
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Abstract

Complementing an earlier paper, which focused on $3$-sumfree sequences, we here consider only $2$-sumfree sequences: starting with positive integers f and g>f, the infinite, increasing 2-sumfree sequence Sf,g is constructed as follows. After any initial segment, the next entry is the smallest positive integer exceeding all previous ones that differs from all sums of distinct pairs in the sequence. It follows from a theorem in the previous paper that for every f≥ 1 and all f+1≤ g<2f the sequence Sf,g exhibits ultimately periodic behaviour. In this paper we state precise conjectures that, if true, would imply that every $2$-sumfree sequence is ultimately periodic. Here periodicity of an increasing sequence is understood to mean periodicity of the sequence of first differences, or, equivalently, of its characteristic sequence. We supply much computational evidence to support the conjectures.

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Cite

BibTeXRIS
BibTeX
@misc{berkel2026periodicity,
  title = {{Periodicity conjectures for all 2-sumfree sequences}},
  author = {Daan van Berkel and Wieb Bosma},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18522},
  eprint = {2609.18522},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18522},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Periodicity conjectures for all 2-sumfree sequences
AU  - Daan van Berkel
AU  - Wieb Bosma
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18522
UR  - https://arxiv.org/abs/2609.18522
AB  - Complementing an earlier paper, which focused on $3$-sumfree sequences, we here consider only $2$-sumfree sequences: starting with positive integers f and g>f, the infinite, increasing 2-sumfree sequence S_f,g is constructed as follows. After any initial segment, the next entry is the smallest positive integer exceeding all previous ones that differs from all sums of distinct pairs in the sequence. It follows from a theorem in the previous paper that for every f≥ 1 and all f+1≤ g<2f the sequence S_f,g exhibits ultimately periodic behaviour. In this paper we state precise conjectures that, if true, would imply that every $2$-sumfree sequence is ultimately periodic. Here periodicity of an increasing sequence is understood to mean periodicity of the sequence of first differences, or, equivalently, of its characteristic sequence. We supply much computational evidence to support the conjectures.
ER  -