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A393377
Minimum Mahler measure of all irreducible and non-cyclotomic quintic polynomials with coefficients in {-1, 0, 1}.
0
1, 3, 4, 9, 7, 1, 6, 1, 0, 4, 6, 6, 9, 6, 9, 5, 8, 6, 5, 2, 9, 8, 2, 4, 0, 4, 4, 0, 8, 7, 3, 1, 1, 5, 7, 5, 7, 4, 0, 6, 7, 0, 6, 6, 9, 6, 0, 9, 2, 6, 4, 4, 4, 9, 9, 0, 1, 6, 2, 0, 1, 0, 6, 4, 7, 2, 0, 5, 8, 6, 0, 7, 8, 9, 8, 7, 7, 7, 5, 5, 2, 4, 9, 3, 4, 8, 3, 7, 1, 6, 4, 8, 3, 2, 3, 4, 0, 2, 1, 6, 3, 4, 9, 8, 8
OFFSET
1,2
COMMENTS
Decimal expansion of the Mahler measure of x^5 - x^4 + x^3 - x + 1 (or x^5 - x^4 + x^2 - x + 1).
All cyclotomic polynomials have Mahler measure = 1.
Minimum Mahler measure M(d) of all non-cyclotomic polynomials degree d with coefficients in {-1, 0, 1} (left column irreducible polynomials, right column reducible):
d=3 1.32471795724474602596090... 1.0000000000000000000000...
d=4 1.3802775690976141156733... 1.3247179572447460259609...
d=5 1.34971610466969586529824... 1.3247179572447460259609...
d=6 1.32471795724474602596090... 1.3247179572447460259609...
d=7 1.37389527572425904577760... 1.3247179572447460259609...
d=8 1.28063815626775759670190... 1.3247179572447460259609...
d=9 1.32471795724474602596090... 1.2806381562677575967019...
d=10 1.17628081825991750654407... 1.2806381562677575967019...
d=11 1.37161288465375962608318... 1.17628081825991750654407...
d=12 1.22778555869459864969600... 1.17628081825991750654407...
Mahler measure 1.17628081825991750654407... (see A073011) is the smallest known value for all polynomials degree at least 40 degree.
Conjecture: Mahler measure M(d) of non-cyclotomic polynomial for every d >= 1.17628081825991750654407...
FORMULA
Root of polynomial 1 - x^3 - x^5 + x^6 - 2*x^7 + x^8 - x^9 + x^10.
EXAMPLE
1.34971610466969586529824...
MATHEMATICA
pol = x^5 - x^4 + x^3 -x + 1; vec = NSolve[pol == 0, x, WorkingPrecision -> 110];
prod = 1; Do[max = Max[1, Abs[x /. vec[[n]]]]; prod = prod max, {n, 1, Length[vec]}]; RealDigits[prod, 10, 105][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Feb 13 2026
STATUS
approved