---

Claim. Theorem 1.1 of A. Dubickas and A. Novikas, No cubic integer polynomial generates a Sidon sequence, Math. Nachr. 294 (2021), no. 10, 1859--1865: if $f(x)=ax^3+bx^2+cx+d\in\mathbb Z[x]$ with $a>0$, then for no $n_0\in\mathbb Z$ is $\{f(n):n=n_0,n_0+1,\ldots\}$ a Sidon sequence. The proof constructs infinitely many solutions of $f(m)+f(n)=f(r)+f(s)$ in pairwise distinct positive integers $m,n,r,s$, through a prime chosen by Dirichlet's theorem and quadratic reciprocity when the discriminant of $f'$ is nonzero, and directly when it is zero. Such a solution gives two pairs $m\ne n$ and $r\ne s$ of nonnegative integers with equal sums $f(m)+f(n)=f(r)+f(s)$, and replacing $f$ by $-f$ covers $a<0$. So no cubic $f$ answers Problem 324: every polynomial with the property has degree at least four. The paper settles Ruzsa's Conjecture 4.2.

Covers. Every cubic $f\in\mathbb Z[x]$. The paper's introduction also proves directly, by explicit families of solutions, that no polynomial of degree one or two works, and recalls that $x^4$ fails; it says nothing about other quartics or about higher degrees.

Depends on. No page of this wiki.

Acceptance. Refereed: Mathematische Nachrichten 294 (2021), no. 10, 1859--1865 (received 7 July 2020, accepted 30 November 2020); the page is dated to the Crossref online publication of 3 October 2021. The site's commentary credits the cubic case to Dubickas and Novikas, but on a problem the site labels OPEN that commentary is not review. The paper's library card records the theorem.