Problem dossier · Number Theory
Lehmer's Mahler measure problem
a.k.a. Lehmer's conjecture
Is there a gap above 1 in Mahler measures of integer polynomials? Lehmer's degree-10 polynomial has measure ≈ 1.176280818; the question is whether any non-cyclotomic integer polynomial does better.
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Status
Open since 1933. Lehmer's polynomial has survived every search; Dobrowolski-type lower bounds are far from the target.
Think you can crack this one? Read the playbook before you announce →
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The Angle of Attack
Compute Mahler measures for exhaustive families of low-degree, low-height polynomials; hunt small measures in special families (Salem numbers, reciprocal polynomials).
Tags: mahler measure · salem numbers · polynomials · heights
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The Lab
No instruments built yet. When this problem gets tackled, its interactive instruments — explorers, searches, verifiers running in the browser — live here. See the Collatz dossier for what a fully tackled problem looks like.
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The Log
Empty. Work on this problem gets logged here as dated entries — constructions tried, code run, dead ends included. Dead ends are results.
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