On $t$-sumfree sequences

Daan van Berkel; Wieb Bosma

REVIEW

On $t$-sumfree sequences

Daan van Berkel, Wieb Bosma

Number Theory · Submitted 15 Sep 2026 · 12 references

Pith's verdict: Solid work on 3-sumfree sequences with a genuine gap: Theorem 23, required for the main result, is stated without proof, which makes Theorem 3 incomplete as written.

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arXiv ID2609.16843 (version 1)

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CitationDaan van Berkel, Wieb Bosma. "On $t$-sumfree sequences." arXiv:2609.16843 (2026). https://pith.science/paper/NEM2XRGH

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Here's my read on van Berkel and Bosma's paper on t-sumfree sequences.

The paper does two real things well. First, it proves eventual periodicity for large infinite families of 2-sumfree and 3-sumfree sequences with explicit moduli and period lengths (Theorems 16, 18, 19). The proof of Theorem 19, the main family for d≥2, is given in detail and looks correct. Second, it provides extensive computational evidence for the conjecture that all 3-sumfree sequences are eventually periodic, reducing the open cases to at most 12 among parameters up to 25. The computational work is honest — they list the hard cases and note that they computed 500,000 terms without detecting periodicity.

Now the soft spot, and it is load-bearing. Theorem 23, which covers the d=1 case of Theorem 3, is stated with the proof entirely omitted: “We leave the (again rather tedious) proof … to the reader.” No sketch, no derivation, no verification. Theorem 3 is billed as the central result for 3-sumfree sequences, and without Theorem 23 it is not proved. The computational evidence does not cover the infinite family of Theorem 23 (only finitely many small parameters are checked). So the paper’s advertised main theorem is incomplete. This is not a minor lacuna; it is a missing proof for a case the authors themselves call nontrivial.

A secondary concern: the claim that 500,000 terms suffice to detect periodicity in the unresolved cases is not backed by any proven bound on preperiod length. The authors acknowledge this implicitly, but a reader deserves an explicit caveat.

That said, the rest of the paper is well-structured, the proofs that are given are clear, and the computational methods are sensible. The authors are honest about what they cannot prove.

Who is this for? Researchers in combinatorial number theory and automatic sequences who care about sumfree sequences and their periodic behavior. The paper advances the state of the art for 3-sumfree sequences significantly, even with the gap.

Should it go to peer review? Yes, but with a firm requirement: the authors must either supply a correct proof of Theorem 23 or restate Theorem 3 without the d=1 case (and mark that case as a conjecture). Without that, the paper overclaims. I would accept to review it, and I would press for the fix.

Referee report

Major objections (2)

[§6, Theorem 23] The proof of Theorem 23 is entirely omitted; the authors state 'We leave the (again rather tedious) proof ... to the reader.' Since Theorem 23 is the d=1 case of the central Theorem 3 (which is stated as a main result in the Introduction), the central claim is unsubstantiated for an infinite family. The paper does not even provide a sketch or outline of the proof. This is a load-bearing gap. The authors must either supply a full proof or explicitly restrict Theorem 3 to d ≥ 2 and adjust all references (title, abstract, Introduction) accordingly.
[§7, Table 2 and Theorem 5] The claim that 'at most 12 exceptions' exist is based on computing 500,000 terms for each triple, with no theoretical guarantee that this exceeds any possible preperiod length. The paper acknowledges this in the reader's weakest assumption (not in the manuscript itself), but the text in §7 does not discuss the possibility that these 12 cases might be periodic with a preperiod longer than 500,000. This computational limitation should be stated explicitly in Section 7, and the wording 'possible exceptions' should be clarified to reflect that periodicity may still hold but could not be proved with the computed segment.

Minor objections (8)

[Abstract] The phrase 'a considerable subclass' is vague; a more precise description (e.g., referencing the parameters of the families) would be helpful.
[§2] The paragraph on Cameron's work and Calkin–Finch is very brief; the connection to the present work could be expanded to help reader understand novelty.
[§3, Lemma 6] The condition (Q - t(t+1)/2)M ≥ (t-1)K is derived from the proof; a short example illustrating how to choose Q and M in practice (e.g., for t=3) would improve readability.
[§4.2] The algorithm description says 'take 8 blocks of the same length ℓ from the end of the computed segment' – it should clarify that ℓ is chosen as a divisor of the total length; the search over ℓ is not explicitly defined.
[§5, Theorem 10] The notation in the congruence description is hard to parse: 'z ≡ {2g+5,2g+7,...,3g+1,4g,4g+2,...} mod 4g+3' – the pattern is not obvious; a more systematic list would help.
[§6, Theorem 19 proof] The proof is written in a narrative style with repeated 'Step 1, Step 2, ...' but the transition to the claimed period and preperiod is somewhat confusing. A more structured format, perhaps with an explicit inductive construction of blocks, would improve verifiability.
[§7, Tables] Table 1 lists cases for d ≤10, but the text says 'results for 11 ≤ d ≤ 25 can be found in the preprint version' – this is not helpful in the manuscript; either include those tables or remove the reference. Also, meaning of k=p=-1 in Table 2 should be explained directly in caption (currently refers to note in text).
[Throughout] Several sentences are missing punctuation (e.g., 'compute the first n terms, where for simplicity of exposition we assume that n is a ninefold n = 9h' should have a comma after 'exposition'). A thorough copyedit is recommended.

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The paper

The paper claims

For a wide family of 3-sumfree sequences, eventual periodicity is proved, leaving only finitely many open cases.

The reading

This paper studies greedy t-sumfree sequences, where each new term is the smallest positive integer not representable as a sum of t distinct earlier terms. For t=3, the authors prove that for every d≥1 and every g large enough relative to f and d, the 3-sumfree sequence S_{f,g,g+d} is eventually periodic. They provide explicit formulas for the preperiod length, period length, and modulus of periodicity. Through extensive computation, they verify that among all triples with starting values up to 25, only 12 remain without a proven period, lending strong support to the conjecture that all such sequences are eventually periodic.

What carries the argument

Lemma 6, a finite-computation criterion for ultimate periodicity: if the characteristic sequence of a t-sumfree sequence contains a sufficiently long run of identical non-null blocks, then the sequence is forced to be periodic. The proof of Theorem 3 uses this lemma together with an explicit description of the first difference sequence for the families (Theorems 19 and 23), showing that after a finite preperiod the differences repeat.

Extended reading notes

Core claim

The central discovery is Theorem 3 (comprising Theorems 19 and 23): for any d ≥ 1, any f ≥ 1, and any g ≥ max(f+1, d-1), the 3-sumfree sequence S_{f,g,g+d} is ultimately periodic. The period and preperiod sizes are expressed in terms of f and g, and the modulus of periodicity is a linear combination of f, g, and d. This result extends earlier work on 2-sumfree sequences and provides a unified framework for understanding periodicity in a large family of 3-sumfree sequences.

Load-bearing premise

For the twelve unresolved cases in Table 2, the claim that periodicity has not occurred within 500,000 terms relies on the unproven assumption that 500,000 terms exceed any possible preperiod length for those parameters.

Editorial extensions

If this is right

All 3-sumfree sequences with second gap d ≥ 1 and g large enough (g ≥ max(f+1, d-1)) are now proven periodic, reducing the open cases to a finite set.
The explicit formulas for preperiod and period allow for efficient generation of these sequences and prediction of their density.
The computational evidence strongly supports the folklore conjecture that every 3-sumfree sequence is eventually periodic.
The methods developed here may be adaptable to t-sumfree sequences for t > 3, potentially leading to similar results.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The twelve unresolved cases may be genuinely aperiodic, or may have preperiods exceeding half a million; a deeper theoretical analysis of these parameters is needed.
The condition g ≥ max(f+1, d-1) in Theorem 3 suggests a threshold beyond which the sequence's behaviour stabilises; exploring the borderline cases could reveal new patterns.
The computational approach using Lemma 6 could be automated to search for periodicities in other t-sumfree families, potentially leading to a classification for all t.

Original abstract

In this paper we explore the world of $t$-sumfree sequences, primarily for $t=2$ and $t=3$. Starting with $t$ increasing positive integers, build an infinite sequence by choosing as the next entry the least positive integer that is not the sum of $t$ distinct previous values. The main open problem concerns the question of the conjectured ultimate periodicity of these sequences. We formulate a sufficient condition to prove that a conjectured preperiod and period are correct. We also prove periodicity for a considerable subclass of 2-sumfree and of 3-sumfree sequences. Finally, we present the results of extensive computations on 3-sumfree sequences.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.46 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.08 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In this paper we explore the world of $t$-sumfree sequences, primarily for $t=2$ and $t=3$. Starting with $t$ increasing positive integers, build an infinite sequence by choosing as the next entry the least positive integer that is not the sum of $t$ distinct previous values. The main open problem concerns the question of the conjectured ultimate periodicity of these sequences. We formulate a sufficient condition to prove that a conjectured preperiod and period are correct. We also prove periodicity for a considerable subclass of 2-sumfree and of 3-sumfree sequences. Finally, we present the results of extensive computations on 3-sumfree sequences.

background contribution method result interpretation main claim

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s0	In this paper we explore the world of $t$-sumfree sequences, primarily for $t=2$ and $t=3$.	contribution 0.58	0.73	0.03	asserted 0.79
s1	Starting with $t$ increasing positive integers, build an infinite sequence by choosing as the next entry the least positive integer that is not the sum of $t$ distinct previous values.	method 0.67	0.21	0.03	asserted 0.87
s2	The main open problem concerns the question of the conjectured ultimate periodicity of these sequences.	gap 0.67	0.08	0.07	speculative 0.44
s3	We formulate a sufficient condition to prove that a conjectured preperiod and period are correct.	contribution 0.88	0.94	0.16	asserted 0.99
s4	We also prove periodicity for a considerable subclass of 2-sumfree and of 3-sumfree sequences.	result 0.90	0.96	0.30	asserted 1.00
s5	Finally, we present the results of extensive computations on 3-sumfree sequences.	contribution 0.69	0.95	0.06	asserted 0.96

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (12)

[1]

Daan van Berkel and Wieb Bosma,Periodicity conjectures for all 2-sumfree sequences, preprint 2026

View on Pith 2026
[2]

Wieb Bosma, René Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuil, Thian Tromp, Using Walnut to Solve Problems from the OEIS, J. Integer Seq.,28(2025), Article 25.3.8

View on Pith 2025
[3]

Wieb Bosma, John Cannon, Catherine Playoust,The Magma algebra system I: The user language, J. Symb. Comput.24(1997), 235-–265

View on Pith 1997
[4]

Neil J. Calkin, Steven R. Finch,Conditions on Periodicity for SumFree Sets, Experimental Mathematics,5:2, (1996), 131–137, DOI: 10.1080/10586458.1996.10504584

View on Pith arXiv 1996
[5]

Rust Team,Rust Programming Language,https://rust-lang.org/, 2012

View on Pith 2012
[6]

P. J. Cameron,Portrait of a typical sum-free set, in: C. Whitehead (ed.),Surveys in Com- binatorics, London Math. Soc. Lecture Notes123, Cambridge: Cambridge University Press, 1987, pp. 13–42

View on Pith 1987
[7]

L.E. Dickson,The converse of Waring’s problem, Bull. Amer. Math. Soc.40(1934), 711–714

View on Pith 1934
[8]

Richard K. Guy,Unsolved Problems in Number Theory, Problem Books in Mathematics, New York: Springer 2004 (third edition)

View on Pith 2004
[9]

Donald Knuth,The Art of Computer Programming4A,Combinatorial Algorithms Part 1, Boston: Addison-Wesley, 2011

View on Pith 2011
[10]

Hamoon Moussavi,Automatic theorem proving in Walnut, (2016), onlinehttps://arxiv. org/abs/1603.06017

View on Pith Pith review arXiv 2016
[11]

Raymond Queneau,Sur les suitess-additives, J. Comin. Theory12(1972), 31–71

View on Pith 1972
[12]

Jeffrey Shallit,The logical approach to automatic sequences: Exploring combinatorics on words with Walnut, London Mathematical Society Lecture Notes Series482, Cambridge University Press, 2022. Email address:daan.v.berkel.1980@gmail.com and bosma@math.ru.nl 14 V AN BERKEL AND BOSMA d (f,g,h) k p m 4 ( 1, 2, 6 ) 7 6 28 5 ( 1, 2, 7 ) 9 8 44 5 ( 1, 3, 8 ) 9 ...

View on Pith 2022

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Daan van Berkel and Wieb Bosma,Periodicity conjectures for all 2-sumfree sequences, preprint 2026

View on Pith 2026

[2] [2]

Wieb Bosma, René Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuil, Thian Tromp, Using Walnut to Solve Problems from the OEIS, J. Integer Seq.,28(2025), Article 25.3.8

View on Pith 2025

[3] [3]

Wieb Bosma, John Cannon, Catherine Playoust,The Magma algebra system I: The user language, J. Symb. Comput.24(1997), 235-–265

View on Pith 1997

[4] [4]

Neil J. Calkin, Steven R. Finch,Conditions on Periodicity for SumFree Sets, Experimental Mathematics,5:2, (1996), 131–137, DOI: 10.1080/10586458.1996.10504584

View on Pith arXiv 1996

[5] [5]

Rust Team,Rust Programming Language,https://rust-lang.org/, 2012

View on Pith 2012

[6] [6]

P. J. Cameron,Portrait of a typical sum-free set, in: C. Whitehead (ed.),Surveys in Com- binatorics, London Math. Soc. Lecture Notes123, Cambridge: Cambridge University Press, 1987, pp. 13–42

View on Pith 1987

[7] [7]

L.E. Dickson,The converse of Waring’s problem, Bull. Amer. Math. Soc.40(1934), 711–714

View on Pith 1934

[8] [8]

Richard K. Guy,Unsolved Problems in Number Theory, Problem Books in Mathematics, New York: Springer 2004 (third edition)

View on Pith 2004

[9] [9]

Donald Knuth,The Art of Computer Programming4A,Combinatorial Algorithms Part 1, Boston: Addison-Wesley, 2011

View on Pith 2011

[10] [10]

Hamoon Moussavi,Automatic theorem proving in Walnut, (2016), onlinehttps://arxiv. org/abs/1603.06017

View on Pith Pith review arXiv 2016

[11] [11]

Raymond Queneau,Sur les suitess-additives, J. Comin. Theory12(1972), 31–71

View on Pith 1972

[12] [12]

Jeffrey Shallit,The logical approach to automatic sequences: Exploring combinatorics on words with Walnut, London Mathematical Society Lecture Notes Series482, Cambridge University Press, 2022. Email address:daan.v.berkel.1980@gmail.com and bosma@math.ru.nl 14 V AN BERKEL AND BOSMA d (f,g,h) k p m 4 ( 1, 2, 6 ) 7 6 28 5 ( 1, 2, 7 ) 9 8 44 5 ( 1, 3, 8 ) 9 ...

View on Pith 2022