Lehmer's conjecture, raised by Derrick Henry Lehmer and also called the Lehmer Mahler measure problem, asserts that there is an absolute constant mu greater than 1 such that every polynomial with integer coefficients either has Mahler measure at least mu or is an integral multiple of a product of cyclotomic polynomials or the monomial x, in which case its Mahler measure equals 1. The smallest known Mahler measure above 1 belongs to a degree ten polynomial known as Lehmer's polynomial, with measure approximately 1.176280818, widely believed to be the true minimal value.
Facts
StatementLehmer's conjecture proposes that there is an absolute constant greater than one such that every polynomial with integer coefficients either has Mahler measure at least that constant or is a product of cyclotomic polynomials and a power of x, meaning every complex root is a root of unity or zero. 1 Progress Toward ResolutionThe conjecture remains unproven. Smyth proved that it holds for every non-reciprocal integer polynomial. Blanksby, Montgomery and Stewart independently proved a lower bound on the Mahler measure in terms of the polynomial's degree, later improved by Dobrowolski and by Voutier in 1996. The smallest known Mahler measure greater than one belongs to Lehmer's polynomial, a degree ten polynomial with Mahler measure approximately 1.176280818, widely believed to be the true minimal value. 1 Prize Status
Prize Status (category)w-freetextdim2b-0926: category derived from a free-text property; original status/verification detail carried on the source property. Classification
Resolution Status Open Questions
Prize StatusNo prize is recorded.
No source consulted this pass names a monetary prize for this conjecture. Connections
In Branch
Source Lehmer's Conjecture (Wikipedia)
Sources
1. Lehmer's Conjecture (Wikipedia)
Wikimedia FoundationLead section
Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer.
Partial results section
Smyth proved that Lehmer's conjecture is true for all polynomials that are not reciprocal, i.e., all polynomials satisfying
In Branch: Number Theory, Lead sentence
Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer.
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