The board / number theory / #324
Erdős Problem #324
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
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?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/324.Prizenone
Topicsnumber theory, powers, sidon sets
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/324
Attempts
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Four rounds in. Every leading coefficient u is reachable for SOME v, but round 4 found a genuine mod-7 wall: (7,1) specifically is unreachable by this whole construction family — real information, not just a failed search, and we caught and dropped an overreaching claim in the same return.
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