The unreasonable rigidity of Ulam sequences

Hinman, Joshua; Kuca, Borys; Schlesinger, Alexander; Sheydvasser, Arseniy

doi:10.1016/j.jnt.2018.06.015

Cited by 5 publications

(15 citation statements)

References 9 publications

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“…The observed empirical fact that for n ≥ 4, the periods λ n grow linearly has been poorly understood up until now. Our goal is to show that this curious phenomenon is deeply tied to the following-seemingly unconnected-numerical observation of Hinman, Kuca, Schlesinger, and Sheydvasser [HKSS19b]. Specifically, they noted that for n ≥ 4, runs of consecutive elements of U (1, n) group into blocks whose endpoints grow linearly.…”

Section: Introduction and Main Resultsmentioning

confidence: 87%

“…Theorem 1.1 (Theorem 1.1 of [HKSS19b]). There exist integer coefficients a i , b i , c i , d i such that for any C > 0, there exists a positive integer N such that for all integers n ≥ N ,…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The original proof of this theorem was nonconstructive and in fact model theoretic in nature, although it was shown in [HKSS19b] that assuming Theorem 1.1, one can prove that there exists an algorithm that will both find the coefficients a i , b i , c i , d i and the minimal integer N 0 that will satisfy the conditions of Theorem 1.1, given C > 0. Our present goal is to give an alternate, constructive proof of Theorem 1.1, based around the properties of algorithms that compute the coefficients a i , b i , c i , d i .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 2 more Smart Citations

The Ulam Sequence of the Integer Polynomial Ring

Sheydvasser

2019

Preprint

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An Ulam sequence U (1, n) is defined as the sequence starting with integers 1, n such that n > 1, and such that every subsequent term is the smallest integer that can be written as the sum of distinct previous terms in exactly one way. This family of sequences is notable for being the subject of several remarkable rigidity conjectures. We introduce an analogous notion of an Ulam sequence in the polynomial ring Z[X], and use it both to give new, constructive proofs of old results as well as producing a new conjecture that implies many of the other existing conjectures.

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Section: Introduction and Main Resultsmentioning

confidence: 87%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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The Ulam Sequence of the Integer Polynomial Ring

Sheydvasser

2019

Preprint

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“…This result, contingent on the veracity of the aforementioned observations, is consistent with the numerical data shown on Figure 1. Hinman, Kuca, Schlesinger, and Sheydvasser's rigidity conjecture states that Ulam sequences U(a, b) follow the same pattern whenever b is in a fixed modulo class of some multiple L of a and b > b 0 for some b 0 ∈ N -a result well grounded in numerical data [HKSS17]. We have looked at (2, 1)-sequences to find similar patterns.…”

Section: Some Empirical Observations About (111)-sequencesmentioning

confidence: 80%

Structures in additive sequences

Kuca

2018

Acta Arith.

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Consider the sequence V(2, n) constructed in a greedy fashion by setting a 1 = 2, a 2 = n and defining a m+1 as the smallest integer larger than am that can be written as the sum of two (not necessarily distinct) earlier terms in exactly one way; the sequence V(2, 3), for example, is given by 3, 4, 5, 9, 10, 11, 16, 22, . . .We prove that if n 5 is odd, then the sequence V(2, n) has exactly two even terms {2, 2n} if and only if n − 1 is not a power of 2. We also show that in this case, V(2, n) eventually becomes a union of arithmetic progressions. If n − 1 is a power of 2, then there is at least one more even term 2n 2 + 2 and we conjecture there are no more even terms. In the proof, we display an interesting connection between V(2, n) and Sierpinski Triangle. We prove several other results, discuss a series of striking phenomena and pose many problems. This relates to existing results of Finch, Schmerl & Spiegel and a classical family of sequences defined by Ulam.2010 Mathematics Subject Classification. 11B13, 11B83.

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“…Daniel Ross studied various aspects of the phenomenon in his 2016 PhD thesis [13]. Hinman, Kuca, Schlesinger & Sheydvasser [7] recently undertook an in-depth study of the Ulam sequence and uncovered several striking new properties as well as partially answering some of the questions raised in the present paper. We also refer the reader to the recent work of Kuca [9] on a related sequence.…”

Section: Ulam Himself Is Not Very Clear About His Motivation and The ...mentioning

confidence: 91%

Ulam Sequences and Ulam Sets

Kravitz,

Steinerberger

2017

Preprint

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The Ulam sequence is given by a 1 = 1, a 2 = 2, and then, for n ≥ 3, the element an is defined as the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives the sequence 1, 2, 3, 4, 6, 8, 11, 13, 16, . . . , which has a mysterious quasi-periodic behavior that is not understood. Ulam's definition naturally extends to higher dimensions: for a set of initial vectors {v 1 , . . . , v k } ⊂ R n , we define a sequence by repeatedly adding the smallest elements that can be uniquely written as the sum of two distinct vectors already in the set. The resulting sets have very rich structure that turns out to be universal for many commuting binary operations. We give examples of different types of behavior, prove several universality results, and describe new unexplained phenomena.

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The unreasonable rigidity of Ulam sequences

Cited by 5 publications

References 9 publications

The Ulam Sequence of the Integer Polynomial Ring

The Ulam Sequence of the Integer Polynomial Ring

Structures in additive sequences

Ulam Sequences and Ulam Sets

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