Record Mahler Measures by Roots Outside

Record Mahler Measures by Roots Outside the Unit Circle

This table lists the polynomial with the smallest known Mahler measure among all irreducible, noncyclotomic polynomials with integer coefficients having a fixed number of roots outside the unit circle.

 D   Measure        Out  Coefficients
 10  1.176280818260   1  1 1 0-1-1-1
 18  1.201396186235   2  1 1 1 0 0-1 0-1 0-1
 18  1.219446875941   3  1 1 1 0-1-1-1 0 0 1
 36  1.229482810173   4  1 1 1 0 0-1-1-1 0 1 1 1 0-1-2-1 0 1 1
 54  1.236566917569   5  1 0 1 0 0 0-1 1-1 1 0 0 1-1 1-1 0 0-1 1-1 1-1 0 0-1 1-1
 54  1.239747974875   6  1 1 1 0 0-1-1-1 0 0 0 0 1 1 1 1 1 0-1-2-2-2-1 0 1 1 1 1
 70  1.242979209676   7  1 0 1 0 0 0-1 0-1 0 0 1 1 1 1 0 0-1-1-1-1 0 0 1 1 1 0 0-1-1-1-1 0 0 1 1
 82  1.244967734300   8  1 0 1 0 0 0-1 0-1 0 0 1 1 1 1 0 0-1-1-1-1 0 0 1 1 1 1 0 0-1-1-1 0 0 1 1 1 1 0 0-1-1
 90  1.246403127917   9  1 0 1 0 0 0-1 0-1 0 0 0 1 1 1 1 0 0-1-1-1-1 0 0 1 1 1 1 0 0-1-1-1-1-1 0 0 1 1 1 1 0 0-1-1-1
 90  1.247685884965  10  1 1 1 0 0-1-1-1 0 0 0 0 0 0 0 1 1 1 1 1 0-1-1-1-1-1 0 0 0 0 1 0 0 0 1 0 0 0 0-1-1 0 0 0 0 1
 98  1.248378407200  11  1 1 1 0 0-1-1-1 0 0 0 0 0 0 0 1 1 1 0 0-1 0 0 1 0 0-1-1-1 0 1 1 0-1-1-1 0 1 2 1 0-1-1-1 0 1 1 0-1-1
118  1.249340932335  12  1 0 1 0 0 0-1 0-1 0 0 0 1 0 1 0 0 1-1 1-1 0 0-1 1-1 1 0 0 1-1 1-1 0 0-1 1-1 1 0 0 1-1 1 0 0 1-1 1-1 0 0-1 1-1 1 0 0 1-1
122  1.249811269316  13  1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-1
132  1.250511624066  14  1 1 1 1 0 0-1-2-2-2-1 0 1 2 3 3 2 1-1-3-4-5-4-2 0 3 5 6 6 4 1-2-5-7-7-6-3 1 4 7 8 7 5 1-3-6-8-8-6-3 1 5 8 9 8 5 1-3-7-9-9-7-3 1 5 8 9
134  1.250935714357  15  1 1 1 0 0-1-1-1 0 0 0 0 0 0 0 1 1 1 0 0-1-1-1 0 0 0 0 1 1 1 1 1 0-1-1-1-1-1 0 0 0 0 1 1 0 0 0 0-1 0 0 0-1 0 0 0 0 1 1 0 0 0 0-1 0 0 0-1
154  1.251434603056  16  1 0 1 0 0 0-1 0-1 0 0 0 1 0 1 0 0 0-1 0-1 0 0 1 1 1 1 0 0-1-1-1-1 0 0 1 1 1 1 0 0-1-1-1-1 0 0 1 1 1 1 0 0-1-1-1 0 0 1 1 1 1 0 0-1-1-1-1 0 0 1 1 1 1 0 0-1-1
150  1.251614950077  17  1 1 1 0 0-1-1-1 0 0 0 0 0 0 0 1 1 1 0 0-1-1-1 0 0 0 0 0 0 0 1 1 1 1 1 0-1-1-1-1-1 0 0 0 0 1 0 0 0 1 0 0 0 0-1-1 0 0 0 0 1 0 0 0 1 0 0 0 0-1-1 0 0 0 0 1
168  1.251995991922  18  1 1 1 1 0 0-1-2-2-2-1 0 1 2 3 3 2 1-1-2-3-4-3-2 0 1 2 3 3 3 1 0-1-2-2-3-2-1 0 1 1 2 2 2 1 0 0-1-1-2-2-1-1 0 0 1 2 2 2 1 1 0-1-2-2-1-1 0 0 1 2 1 1 0 0 0-1-1-1 0 0 0 0 0 1
166  1.252179060410  19  1 1 1 0 0-1-1-1 0 0 0 0 0 0 0 1 1 1 0 0-1-1-1 0 0 0 0 0 0 0 1 1 1 0 0-1-1-1 0 0 0-1-1-1 0 1 1 1 0 0-1 0 0 1 0 0-1-1-1 0 1 1 1 0 0-1 0 0 1 0 0-1-1-1 0 1 1 1 0 0-1 0 0 1
164  1.252712341185  20  1 1 1 0 0-1-1-1 0 0 0 0 0 0 0 1 1 1 0 0-1-1-1 0 0 0 0 0 0-1 0 0 1 0 1 0 0-1 0 0 0 0 0 0-1 0 0 1 0 1 0 0-1 0 0 0 0 1 1 0 0 0 0-1 0 0 0-1 0 0 0 0 1 1 0 0 0 0-1 0 0 0-1
174  1.253366478536  21  1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-1
136  1.282679391168  22  1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 1
142  1.282867471862  23  1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1-1
148  1.283037021353  24  1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 1
154  1.283190498494  25  1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1-1
160  1.283329958038  26  1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 1
166  1.283457128942  27  1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1-1
168  1.283573475346  28  1 0-1 1 0-2 1 1-2 1 2-2 0 2-2-1 2-1-1 2 0-1 1 0-1 0 0 0 0 0 1 0-1 1 0-2 1 1-2 1 2-2 0 2-2-1 2-1-1 2 0-1 1 0-1 0 0 0 0 0 1 0-1 1 0-2 1 1-2 1 2-2 0 2-2-1 2-1-1 2 0-1 1 0-1
178  1.283680244512  29  1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1 0 1 1 0-1-1-1
178  1.284913043305  30  1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-1
178  1.285626299068  31  1 1 0 0 0 -1 -1 0 0 0 1 1 0 0 0 -1 -1 0 0 0 1 1 0 0 0 -1 -1 0 0 0 1 1 0 0 0 -1 -1 0 0 0 1 1 0 0 0 -1 -1 0 0 0 1 1 0 0 0 -1 -1 0 0 0 1 1 0 0 0 -1 -1 0 0 0 1 1 0 0 0 -1 -1 0 0 -1 0 1 0 0 1 0 -1 0 0 -1

History

December 20, 1996: Original list published for values of Out up to 30.
January 31, 2000: Added entry for Out=31 (found by Petr Lisonek).

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Last modified June 5, 2009.