A criterion for sign alternation along parent chains in Ulam sequences
Year: 2026
Type: preprint
Abstract: Every term n > 2 of the Ulam sequence 1, 2, 3, 4, 6, 8, 11, … is the sum of two distinct earlier terms in exactly one way, n = a(n) + p(n) with a(n) < p(n). Let α ≈ 2.571447 be Steinerberger's frequency and let r(x) ∈ (−π, π] be αx reduced modulo 2π. For every term n with 3 < n ≤ 2^28 (all terms not exceeding 2^28 ≈ 2.7·10^8) except n = 4 and n = 8, the residues r(a(n)) and r(a(p(n))) have opposite signs; this is a finite computation with the fixed rational value α = 2πP/2^40, P = 449984567789, the dyadic rational nearest to Gibbs's estimate. We give a short elementary explanation: if the angles of n, p(n) and p(p(n)) lie in an arc of length w < π and the two residues have the same sign, then |r(a(n))| + |r(a(p(n)))| ≤ w, while uniqueness forces a(n) + a(p(n)) not to be a term, apart from two explicit exceptions (the case n = 8 is one of them). The argument uses only the uniqueness of the representation, so it applies to any α and any 1-additive sequence, and it reduces sign alternation in a given range to a finite check. We carry out this check in exact integer arithmetic for the coprime initial pairs (c, d) with c < d ≤ 20, using the terms up to 10^6 and a frequency estimated from them: in this setting 54 of the 127 pairs have an arc shorter than a half-turn, and the check succeeds only for (1, 2) and (1, 3); for the other 52, alternation does fail, and always in the way the argument allows. The arc hypothesis is an empirical observation going back to Steinerberger and Gibbs and is not proved here. The proofs are short and written out in full, 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: W7225056604