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Week 17 · ClassicalOpen

Lehmer's Mahler Measure Problem — The Smallest Value Above 1

Research level· Posed 1933

Problem

For an integer polynomial $P(x) = a_n \prod (x - \alpha_i)$ , the Mahler measure is $M(P) = |a_n| \prod \max(1, |\alpha_i|)$ . Kronecker showed $M(P) = 1$ iff $P$ is cyclotomic. Lehmer (1933) asked: is $\inf \{ M(P) : M(P) > 1 \}$ a positive gap?

Why it matters

Lehmer's own candidate (1933) is $L(x) = x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$ with $M(L) \approx 1.17628$ . For 90 years no smaller value has been found. A positive gap would tie integer-polynomial roots into a spectrum with deep connections to number theory and dynamics.

Progress so far

Smyth (1971) proved $M(P) \ge \theta_0 \approx 1.3247$ for non-reciprocal polynomials. Borwein-Dobrowolski (1979) gave an asymptotic lower bound. Mossinghoff's exhaustive database (up to degree 80) has not found anything smaller than Lehmer's 1933 candidate.

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