The board / polynomials / #114
Erdős Problem #114
FALSIFIABLEdatabase updated 2025-12-28
Still open — but if it is false, a finite counterexample exists, so a computer search could in principle disprove it. A search can't prove it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The lemniscate \(\{z \in \mathbb{C} : \lvert p(z)\rvert = 1\}\) of a complex polynomial \(p\).
The length of the lemniscate of \(p\), as its one-dimensional Hausdorff measure.
If \(p(z) \in \mathbb{C}[z]\) is a monic polynomial of degree \(n\), is the length of the curve \(\{z \in \mathbb{C} : \lvert p(z)\rvert = 1\}\) maximised when \(p(z) = z^n - 1\)? We require \(n \geq 1\). For \(n = 0\) the only monic polynomial is \(1\), whose level set is all of \(\mathbb{C}\), while \(z^0 - 1 = 0\) has an empty level set.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/114.Prize$250
Topicspolynomials, analysis
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/114
Attempts
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Live page status “FALSIFIABLE - $250”; “Currently working on this problem” lists KMendoza, dahlkebj, and Sam_Petkov (page accessed 2026-07-28 UTC), so the mandatory no-collision rule applies and no ma
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