The Ulam Sequence of the Integer Polynomial Ring

TL;DR

This paper extends the concept of Ulam sequences to the polynomial ring Z[X], providing new proofs of known results and proposing a conjecture that links several existing conjectures in the field.

Contribution

It introduces an Ulam sequence framework within Z[X], offering novel constructive proofs and a unifying conjecture for multiple open problems.

Findings

New Ulam sequence definitions in Z[X]

Constructive proofs of classical results

A new conjecture linking existing conjectures

Abstract

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 inside 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.