MATHEMATICS OF COMPUTATION Volume 67, Number 224, October 1998, Pages S11–S14 S 0025-5718(98)01006-0

Supplement to POLYNOMIALS WITH SMALL MAHLER MEASURE

MICHAEL J. MOSSINGHOFF

Table 1. Mahler measures less than 1.24

d Measure  ν Half of coefficients
10 1.176280818260 1 1 1 0−1−1−1
18 1.188368147508 1 1 1 1 1 0 0−1−1−1−1
14 1.200026523987 1 1 0 0 1−1 0 0−1
18 1.201396186235 2 1 1 1 0 0−1 0−1 0−1
14 1.202616743689 1 1 0−1 0 0 0 0 1
22 1.205019854225 2 1 0 1 0 0 1−1 1 0 0 1−1
28 1.207950028412 2 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1
20 1.212824180989 2 1 1 0 0 1 1 0−1−1−1−1
20 1.214995700776 2 1 1 1 1 1 0−1−1−1−1−1
10 1.216391661138 1 1 0 0 0−1 1
20 1.218396362520 2 1 1 0−1 0 0 0−1 0 0 1
24 1.218855150304 2 1 0 0 0 0 1 0−1 0 0 0 0−1
24 1.219057507826 2 1 1 0−1−1−1−1−1 0 1 1 1 1
18 1.219446875941 3 1 1 1 0−1−1−1 0 0 1
18 1.219720859040 1 1 1 0 0 0 0 0 0−1−1
34 1.220287441693 3 1 0 1 0 0 1−1 1−1 0 0−1 1−1 0 0−1 1
38 1.223447381419 3 1 1 0−1−1 0 0−1−1 0 1 1 0 0 1 1 0−1−1−1
26 1.223777454948 3 1 1 1 0 0−1−1−1−1 0 0 1 0 1
16 1.224278907222 2 1 1 0−1−1 0 1 1 1
18 1.225503424104 2 1 1 0 0 1 0−1 0 0−1
30 1.225619851977 2 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1
30 1.225810532354 3 1 0 1 0 0 0−1 1−1 1−1 0 0−1 1−1
26 1.226092894512 2 1 0 0 1 0 0 0 0 0 0 1 0 0 1
36 1.226493301473 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0−1−1
20 1.226993758166 2 1 1−1−2 0 2 1−1−1 0 1
12 1.227785558695 2 1 1 1 0−1−1−1
30 1.228140772740 3 1 1 0 0 0−1−1−1−1 0 0 0 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
22 1.229566456617 3 1 1 0−1−1−1−1−1 0 1 1 1
34 1.229999039697 3 1 0 0 0 0 0−1 0 0 0 0 1 0 0 0 0 0−1
38 1.230263271363 3 1 0 0 0−1 0 0 0 0 0 0 0 0 0 0 1 0 0 0−1
42 1.230295468643 4 1 0 1 0 0 0−1 1−1 1 0 0 1−1 1−1 1 0 0 1−1 1
10 1.230391434407 1 1 0 0 1 0 1
46 1.230743009076 3 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0−1 0 0−1 0−1 0 0−1
18 1.231342769993 3 1 0 0 0−1 1 0 0 0−1
48 1.232202952743 4 1 1 0 0 1 1 0−1−1 0 0−1−1 0 1 1 0 0 1 1 0−1−1−1−1
20 1.232613548593 1 1 1 0 0 0 1 1 0 0 1 1
28 1.232628775929 2 1 2 2 1 1 1 0−1−1−1−1−1−1−1−1
38 1.233672001767 3 1 1 0−1−1−1−1 0 1 1 0 0 0 0 0 1 1 0−1−1
52 1.234348374876 4 1 1 0−1−1 0 0−1−1 0 1 1 0 0 1 1 0−1−1 0 0−1−1 0 1 1 1
24 1.234443834873 2 1 0 0 0−1 0−1 1 0 0 1−1 1
26 1.234500336789 3 1 0−1 1 0−1 1 0−1 1 0−1 0 1
16 1.235256705642 2 1 1 0 0 0−1 0 0−1
46 1.235496042193 3 1 0−1 0 0 0 0 1 0−1 0 0 0 0 1 0−1 0 0 0 0 1 0−1
22 1.235664580390 1 1 0−1 1 0 0 0−1 1 0−1 1
42 1.235761099712 4 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1
32 1.236083368052 4 1 1 1 1 0−1−1−2−1 0 0 1 1 0 0 0−1
32 1.236198469859 3 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1
32 1.236227922245 4 1 2 2 1 0−1−1−1−1−1−1−1−1−1 0 2 3
40 1.236249557349 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1
16 1.236317931803 1 1 1 0 0 0 0 0 0−1
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
34 1.236579223637 3 1 0 0 0 0 1 0 0 0 0 0 0−1 0 0 0 0−1
44 1.236674812187 4 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0−1 0 0 0 0 0 0−1
28 1.236808305865 2 1 1 0−1 0 0 0 0 0−1 0 0 0 0 1
26 1.237504821217 1 1 0−1 0 0 1 0 0−1 0 1 0 0−1
46 1.237634830280 5 1 1 0 0 0 0 0−1−1 0 0−1−1 0 1 0−1 0 1 1 0−1 0 1
58 1.237684127894 5 1 1 0 0 0−1−1−1−1 0 0 0 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1 0 0 1
48 1.238040100176 4 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0−1 0 0 0 0−1
56 1.238431627359 4 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1
56 1.238708978554 4 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0−1 0 0−1
42 1.239505770490 4 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 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
48 1.239861326360 4 1 2 3 3 2 0−2−3−3−2−1 0 0−1−2−2−1 0 1 1 1 1 1 1 1

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

S11 S12 SUPPLEMENT

Table 2. Smallest known Mahler measures by degree (minimal Mahler measure for d ≤ 24)

d Measure  ν Half of coefficients
8 1.280638156268 1 1 0 0 1−1
10 1.176280818260 1 1 1 0−1−1−1
12 1.227785558695 2 1 1 1 0−1−1−1
14 1.200026523987 1 1 0 0 1−1 0 0−1
16 1.224278907222 2 1 1 0−1−1 0 1 1 1
18 1.188368147508 1 1 1 1 1 0 0−1−1−1−1
20 1.212824180989 2 1 1 0 0 1 1 0−1−1−1−1
22 1.205019854225 2 1 0 1 0 0 1−1 1 0 0 1−1
24 1.218855150304 2 1 0 0 0 0 1 0−1 0 0 0 0−1
26 1.223777454948 3 1 1 1 0 0−1−1−1−1 0 0 1 0 1
28 1.207950028412 2 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1
30 1.225619851977 2 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1
32 1.236083368052 4 1 1 1 1 0−1−1−2−1 0 0 1 1 0 0 0−1
34 1.220287441693 3 1 0 1 0 0 1−1 1−1 0 0−1 1−1 0 0−1 1
36 1.226493301473 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0−1−1
38 1.223447381419 3 1 1 0−1−1 0 0−1−1 0 1 1 0 0 1 1 0−1−1−1
40 1.236249557349 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1
42 1.230295468643 4 1 0 1 0 0 0−1 1−1 1 0 0 1−1 1−1 1 0 0 1−1 1
44 1.236674812187 4 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0−1 0 0 0 0 0 0−1
46 1.230743009076 3 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0−1 0 0−1 0−1 0 0−1
48 1.232202952743 4 1 1 0 0 1 1 0−1−1 0 0−1−1 0 1 1 0 0 1 1 0−1−1−1−1
50 1.240379074717 3 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0−1−1
52 1.234348374876 4 1 1 0−1−1 0 0−1−1 0 1 1 0 0 1 1 0−1−1 0 0−1−1 0 1 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
56 1.238431627359 4 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1
58 1.237684127894 5 1 1 0 0 0−1−1−1−1 0 0 0 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1 0 0 1
60 1.240061859037 4 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 0 0 1
62 1.244598861428 6 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
64 1.246918808403 4 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 0−1−1
66 1.245397804852 7 1 0 1 0 0 0−1 0−1 0 0 0 1 1 1 1 0 0−1−1−1−1−1 0 0 1 1 1 1 0 0−1−1−1
68 1.241372531335 6 1 1 0 0 0 0 1 0−1 0 0 0 0−1 0 1 0 0 0 0 1 0−1 0 0 0 0−1 0 1 0−1−1 0 1
70 1.241422024928 5 1 1 0−1−1−1−1 0 1 1 0 0 0 0 0 1 1 0−1−1−1−1 0 1 1 0 0 0 0 0 1 1 0−1−1−1
72 1.241788568356 6 1 1 0 0 0−1−1−1−1 0 0 0 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1 1 1 1 0 0 0−1 0 0−1
74 1.246055240158 7 1 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 1 0 0 0 0 0 0 0 0 0−1
76 1.242775741593 5 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 0 0 1 0 0 1 0 1 0 0 1
78 1.245376549311 8 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 1 0 0−1−1−1−1 0 0 1 1 1
80 1.245374741297 6 1 1 0−1−1 0 0−1−1 0 1 1 0 0 1 1 0−1−1 0 0−1−1 0 1 1 0 0 1 1 0−1−1 0 0−1−1 0 1 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
84 1.247632944895 8 1 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 0 0 0 0 0 0 0 0 0 0−1
86 1.244953592390 7 1 1 0 0 0−1−1−1−1 0 0 0 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1 0 0 1
88 1.245922202165 8 1 1 0 0 0 0 0−1 0 1 0 0 0 0 1 0−1 0 0 0 0−1 0 1 0 0 0 0 1 0−1 0 0 0 0−1 0 1 0−1−1 0 1 0−1
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
92 1.252142335381 8 1 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 0 0 0 0 0 0 0 0 0 0 0 0−1 0 0 0 0 0 0−1
94 1.247133538468 9 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
96 1.247821565271 10 1 1 1 1 0 0−1−2−2−2−1 0 1 1 2 2 1 1−1−1−1−2−1−1 0 1 1 1 1 1 0 0−1−1 0−1 0 0 0 1 1 1 1 1 0 0−1−2−1
98 1.247395763109 9 1 1 0 0 0 0 0−1−1−1−1 0 0 0 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1 0 0 1 0 0 0−1
100 1.247365098370 8 1 1 0 0 0−1−1−1−1 0 0 0 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1 1 1 1 0 0 0−1−1−1−1 0 0 0 1 1 1 1 0 0 0−1 0 0−1

SUPPLEMENT S13

Table 3. Number of primitive, irreducible polynomials found with measure < 1.3 classified by degree and the number of roots outside the unit disk

d 1  2  3  4  5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Total
8 1  1
10 5  2  7
12 1  4  5
14 4  5  2  11
16 1 12  1  14
18 7  8  8  23
20 4 22  3  6  35
22 4 12 27  4  1  48
24 2 29  3 11  1  46
26 7 11 21 11 11  61
28 1 15 14 28  1  59
30 4 10 36 20  8 2  80
32  12  8 35  7 7  69
34 2  5 24 19 30 8 2  90
36 1 15 11 36  4 20  1  88
38  1 18 18 29 8 12  86
40 1  9  7 32 10 23 2 3  87
42  11 14 16 6 8 2  57
44 1  2  4 29  5 25 4 7  77
46 1  1 10 13 36 11 23 7 3  105
48  2  4 24  4 24 1 19 1  79
50  4 10 15 14 16 9 8  76
52  2  7  8 21 4 16 4 3  65
54  4 14 15 14 3 15 1  66
56  6  5 16 7 11 4 6  55
58  3  1 15 12 21 9 25 9 4  99
60  2  5  4 19 4 13  18  65
62  1  10 12 20 7 19 9 13  91
64  7  2 8 6 17 5 16 5 3  69
66  1  5 5 11 11 7  6  46
68  1  1  2 11 11 12 3 12 5 10  68
70  7 9 24 16 21 7 26 7 5  122
72  1  1 10 2 18  8  17  57
74  3 4 15 12 15 6 17 10 10  92
76  1  2 7 9 13 8 11 3 18 5 3  80
78  2 4 10 12 15  12 2 13 2  72
80  3  6 4 16 4 11 2 12 7 10  75
82  2 1 11 11 21 12 16 8 30 11 3  126
84  1 5 3 10 4 12 1 7  18  61
86  2  1 2 11 7 22 11 10 6 19 10 16  117
88  2 2 13 4 16 6 10 4 17 4 3  81
90  1 6 3 13 11 12 2 8 1 11 2  70
92  1 2 4 6 9 5 10 3 15 7 9  71
94  6 9 14 14 25 7 14 9 34 10 5  147
96  4 2 11 3 16 2 10  6  18  1  73
98  1  3 4 12 10 18 10 10 6 15 10 11  110
100  1 1 1 4 7 10 9 14 2 13 6 25 4 4  101
102  3 6 8 5 17 8 5 2 13 2 16 2  87
104  1  5 3 8 5 14 4 5 4 12 6 10  77
106  1  2 4 11 8 19 12 18 5 14 8 28 8 2  140
Total 47 177 225 349 264

S14 SUPPLEMENT

Table 3. (Continued)

d  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Total
108  1  1  2  1 11  2 13  4 11  8  22  76
110  2  2 10  4 15 14 16  4 13  7 21 10 18  136
112  4  3 11  4 14  5 10  2  8  6 19  4  4  94
114  1  2  7  5  9 10 17  6  5  1  7  2 13  2  87
116  1  4  2  4  5 10  7 11  3  7  3 13  8  8  86
118  6  6 18 10 27 11 17  7 16  9 34 13  8  182
120  4  1 10  2 10  3 14  11  13  26  1  95
122  1  4  4 10  8 19 11 11  5  9 10 19 12 11  134
124  2  4  4  4  9 15  7 10  4 10  4 22  6  2  103
126  1  4  5  8  4 11 13 13  6  8  12  85
128  1  1  2  3  9  4 14  6  9  2  7  6 12  8 14  98
130  6  2 13 10 17 14 20  6 14  6 15 10 36 12  3  184
132  4  3  8  2 14  5 10  10  14  23  93
134  1  4  7  6 14  9 24 10  8  5 15  7 22 14 20  166
136  1  1  4  3 10  4 10  6 18  4  4  3  7  6 19  5  1  106
138  2  1  3  8 10  7 17  7  7  2  5  2  9  1 15  3  99
140  1  4  1  7  4  8  7 12  3  8  3  8  3 15  6 14  104
142  2  9  7 13  9 24 19 23  3 14  7 19 13 41 12  5  220
144  3  1  8  1  9  3 17  2  9  8  12  22  1  96
146  1  4  2 14  8 12 16 21  5  9  5 11  7 18 15 14  162
148  2  3  6  8  6 12 10  6  3 11  3 13  6 27  6  4  126
150  3  5  2 11  4 10  9 19  6  3  8  1 11  2 19  2  115
152  1  2  5  2  8  4  8  6 14  3  7  3  5  5 11  8 13  105
154  5  5  9  9 16 10 23 15 11  4 14  7 14  8 35 12  4  201
156  1  1  4  2  7  2 14  5 13  1  7  9  14  24  104
158  3  3  9  6 13 10 22 14 17  5 13  5 14  9 25 14 22  204
160  1  4  1 13  3 10  6 20  5  9  1 10  4  9  4 27  6  4  137
162  3  3  4  6  8  4 13 13 14  2  4  1  8  2 10  2 16  3  116
164  3  3  2  4  9  6 14  7 10  3  6  3  9  5 13  9 12  118
166  1  2 10  6 15  7 18 14 30  8 14  5 12  6 18 11 39 10 5  231
168  1  5  1  7  2 11  2 17  4 14  10  7  14  29  1 125
170  1  3  4  5 10  6 13 10 20  7  7  5  7  5 12  8 21 14 15  173
172  1  1  1  6  5  9  6  8 12 15  3  8  4 10  3 13  6 29 7 3 150
174  3  3 10  6 11  7 17 12 11  1  8  1  9  2  9  2 23 3 138
176  2  2  1  3  6  3  6  4 10  6  4  2  3  3  6  6 10 8 8  93
178  1  4  5  8  6 12  9 17 14 16  3  9  6 10  6 18 11 34 11 200
180  1  1  2  1  2  2  5  3  7  2  2  2  3  5  7  45
Total 329 287 345 337 355 388 416 431 439 442 426 403 383 351 309 259 228 215 198 186 175 156 129 92 33 8374