Problem 324

Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a<b$ nonnegative integers are distinct?

Status

Open

No claim settles this problem.

Open: the site's label; its commentary credits Dubickas and Novikas's exclusion of cubic polynomials (claim page (Dubickas and Novikas, 2021)).

DuNo21: Dubickas, Arturas and Novikas, Aivaras, No cubic integer polynomial generates a Sidon sequence. Math. Nachr. 294 (2021), 1859-1865.
Gu04: Guy, Richard K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section F30 "A polynomial whose sums of pairs of values are all distinct", printed p. 403, which states the problem as Erdős's and names $x^5$ as "a likely answer", with Ruzsa's almost polynomial Sidon set [Ru01b]. Library home: Guy 2004.
Ru01b: Ruzsa, I. Z., An almost polynomial Sidon sequence. Studia Sci. Math. Hungar. (2001), 367-375.