Ultimate periodicity of 2-sumfree sequences

For every pair of integers $f\ge 1$ and $g>f$ , define the increasing sequence $S_{f,g}=(a_n)_{n\ge 1}$ by $a_1=f$ , $a_2=g$ , and, recursively, $a_{n+1}$ is the smallest integer greater than $a_n$ that is not the sum of two distinct earlier terms; that is, $a_{n+1}=\min\{m>a_n:m\ne a_i+a_j\text{ for all }1\le i<j\le n\}$ . Then the sequence of first differences $d_n=a_{n+1}-a_n$ is ultimately periodic: there exist integers $N\ge 1$ and $p\ge 1$ such that $d_{n+p}=d_n$ for every $n\ge N$ .

References

Primary source

arXiv

Additional references

Periodicity conjectures for all 2-sumfree sequences — arXiv — Daan van Berkel, Wieb Bosma

additive-combinatorics computational-mathematics

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A new paper extends conjectures and computations to all such sequences, but does not prove that every sequence eventually repeats.

The problem asks whether every greedy $2$ -sumfree sequence eventually repeats, which would classify all such sequences uniformly. The current work is by Daan van Berkel and Wieb Bosma.

September 2026 conjectural extension

A September 16, 2026 report on Periodicity conjectures for all $2$ -sumfree sequences gives period and preperiod formulas for the remaining parameter range and supplies computational evidence. In On $t$ -sumfree sequences, submitted September 15, 2026, van Berkel and Bosma prove periodicity for a considerable subclass and give sufficient conditions for the conjectured formulas, but explicitly do not prove ultimate periodicity for all $2$ -sumfree sequences.

Current status (as of September 2026): ultimate periodicity for all $2$ -sumfree sequences remains open; only a substantial subclass and conditional or computational evidence are reported.

Sources

No solutions have been posted yet.