Lehmer's conjecture
Proposed lower bound on the Mahler measure for polynomials with integer coefficients / From Wikipedia, the free encyclopedia
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For Lehmer's conjecture about the non-vanishing of τ(n), see Ramanujan's tau function. For Lehmer's conjecture about Euler's totient function, see Lehmer's totient problem.
Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer.[1] The conjecture asserts that there is an absolute constant such that every polynomial with integer coefficients satisfies one of the following properties:
- The Mahler measure[2] of is greater than or equal to .
- is an integral multiple of a product of cyclotomic polynomials or the monomial , in which case . (Equivalently, every complex root of is a root of unity or zero.)
There are a number of definitions of the Mahler measure, one of which is to factor over as
and then set
The smallest known Mahler measure (greater than 1) is for "Lehmer's polynomial"
for which the Mahler measure is the Salem number[3]
It is widely believed that this example represents the true minimal value: that is, in Lehmer's conjecture.[4][5]