Long gaps, twin pairs and forbidden differences in the Ulam sequence
Year: 2026
Type: preprint
Abstract: Erdős problem #342 asks three questions about the Ulam sequence 1, 2, 3, 4, 6, 8, 11, …; one of them is whether there are infinitely many pairs a, a + 2 of terms (twin pairs). We study the differences of consecutive terms through the angles {nβ}, β = α/2π, where α is Steinerberger's frequency. We assume an explicit hypothesis (H): all terms other than 2, 3, 47, 69, 102 lie within 26.6° of the middle third. Under (H), a term whose angle lies in [213.3°, 240°) is always followed by a term two steps later, unless that number is a sum of two outliers. It follows that the number of pairs a, a + 2 up to N is at least about |U ∩ [1, N]|/(52(|E_N| + 1)), minus corrections that count sums of two outliers. So infinitely many pairs would follow from a growth condition on the number |E_N| of outliers. For the differences that OEIS A072540 lists as not occurring, short angle arguments, some of which go back to Gibbs and McCranie, give the following. Under (H), 11 and 33 never occur. Unconditionally, 6, 16, 18, 21, 23, 26, 28 cannot occur between two terms of the middle third. Under (H), 14 and 31 can occur there only when n + 2 is a sum of two outliers. With (H) alone, a finite case analysis of everything visible from the start of a gap adds only one further value below 1500. With the exact list of the outliers below 10^4 and a 4.7° version of (H), the same analysis excludes, as differences between two terms of the middle third, all but 17 of the 173 values below 300 that do not occur up to 2^31; the two smallest survivors are 54 and 102. Up to 2^31, all 1.6·10^8 differences between terms of the middle third agree with both analyses. Without hypotheses we describe the long gaps. In the computed data their lengths lie on the lattice 521 or 530 plus 22a + 259b with |a| ≤ 6, built from denominators of convergents of β, and every record gap of length at least 262 in the record list up to about 10^12 lies on it. One more turn of 259 steps costs a factor of about 1/40 to 1/49. A formula built from the outliers and the angular distribution gives 1/36, and two measured corrections account for the rest. Apart from an elementary angle statement, nothing is proved about the Ulam sequence without hypotheses. The arguments are written out in full and the finite case analyses are carried out with the deposited code, but their validity is left open to external verification; see the note at the end of the paper. The accompanying archive contains the programs and the recorded outputs.
Author: Funayama, Shunsuke
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225040941