For every integer t≥ 2 and every strictly increasing t-tuple of positive integers a_1<…<a_t, define a set recursively by A_0=a_1,…,a_t and, for each n≥ 0, adjoining the least positive integer that is neither already in A_n nor expressible as a sum of t distinct elements of A_n. The conjecture is that every resulting set A=bigcup_n≥ 0A_n is ultimately periodic: there exist integers N≥ 1 and p≥ 1 such that, for every integer m≥ N, m∈ A if and only if m+p∈ A.

Ultimate periodicity of t-sumfree sequences

The problem asks whether recursively defined $t$ -sumfree sequences are ultimately periodic, especially in the unresolved $2$ -sumfree and $3$ -sumfree cases. Cameron’s $1987$ question isolates a related implication between periodicity and regularity.

Known results

Every finite-greedy sum-free sequence is regular (Calkin and Finch, $1996$ ).
A general period bound is $2^{a_L}+1$ ; sharper bounds are $2a_1$ for $L=1$ and $a_1+a_2$ for $L=2$ (Calkin and Finch, $1996$ ).
A positive linear lower bound on representation counts outside an infinite sum-free set implies regularity.

Recent subclass results, September 2026

A September $15$ , $2026$ update reports periodicity for substantial subclasses and a verification criterion for proposed preperiods and periods (van Berkel and Bosma). Separately, a result for the greedy $3$ -sumfree sequence $S_{1,g,g+1}$ proves eventual periodicity for every $g\geq 2$ , with residue period $10g+3$ and block length $2g+1$ . These results do not establish the universal conjectures.

Current status (as of September 2026): substantial subclasses and verification criteria are established, but ultimate periodicity for general $t$ -sumfree sequences, including the universal $2$ -sumfree and $3$ -sumfree conjectures, remains open.

Ultimate periodicity of t-sumfree sequences

Primary source

Additional references

Progress summary

Solutions 0