##
**
The Ulam Sequence of Linear Integer Polynomials
**

###
Arseniy (Senia) Sheydvasser

Israel Institute of Technology

Department of Mathematics

Haifa 3200003

Israel

**Abstract:**

The 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 integer polynomial
ring, and use it both to give new, constructive proofs of old results
as well as to produce a new conjecture that implies many of the other
existing conjectures.

**
Full version: pdf,
dvi,
ps,
latex
**

(Concerned with sequences
A002858.)

Received July 11 2021; revised version received November 30 2021.
Published in *Journal of Integer Sequences*,
December 27 2021.

Return to
**Journal of Integer Sequences home page**