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Updated Oct 7

Problem 324

claims/: The 2 claim pages of Problem 324, one per claimant's result; the problem's standing derives from them.

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

Source. erdosproblems.com/324, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #324, https://www.erdosproblems.com/324.

References.

[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_unsolved_problems_number_theory.
[Ru01b] Ruzsa, I. Z., An almost polynomial Sidon sequence. Studia Sci. Math. Hungar. (2001), 367-375.

Formalization. Statement in formal-conjectures.

Current assessment

No polynomial of degree at most three works. Degrees one and two fail by explicit families of colliding sums (the site calls the quadratic case easy to check; Dubickas and Novikas give the families in their introduction), and the cubic case is Theorem 1.1 of Dubickas and Novikas [DuNo21], a refereed result on its claim page. The site's commentary calls the failure of $x^4$ classical; Euler's identity $59^4+158^4=133^4+134^4$ is a collision. Collin Yuanjie Ren's Lean proof of the degree-at-most-two and $x^4$ cases, prepared with Claude Code, is a pending claim on its page. Ruzsa's set $\{n^5+\lfloor cn^4\rfloor:n\ge n_0\}$ [Ru01b] is a Sidon set for some $c\in[0,1]$ but is not the value set of a polynomial, so it settles no instance. The site expects $f(x)=x^5$ to work, and notes that the Lander, Parkin and Selfridge conjecture would give the property for $x^n$ with every $n\ge5$ . The sources cited here decide no polynomial of degree four or more other than $x^4$ .

Linked library material

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Linked from (10)

Dubickas and Novikas exclude cubic polynomials Ren's Lean proof of the low-degree and quartic cases problems/diophantine_problems/E0324/claims Diophantine Problems and Powers Diophantine Problems and Powers diophantine_problems/dubickas_2021_no_cubic_integer_polynomial_generates_sidon Lemma 2.1 (p. 1861): a prime p with 4B dividing p+1 and -A a quadratic residue mod p Theorem 1.1 (p. 1860): the values of a cubic integer polynomial from any starting point are never a Sidon sequence Unnumbered observation (p. 1860): linear and quadratic integer polynomials have explicit colliding pair sums number_theory/guy_2004_unsolved_problems_number_theory

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