3062:
2365:
1625:
1141:
651:
521:
1014:
3304:
3179:
793:
2255:
2512:
1739:
2829:
859:
2589:
3481:
3402:
3661:, this means that the moduli space of all ergodic compact group automorphisms up to measurable isomorphism is either countable or uncountable depending on the solution to Lehmer's problem.
1210:
2096:
3649:
automorphisms of finite positive entropy or only has automorphisms of infinite entropy depending on the solution to Lehmer's problem. Since an ergodic compact group automorphism is
2177:
891:
320:
2005:
172:
123:
2640:
2873:
1270:
3613:
of a polynomial with integer coefficients if it is finite. As pointed out by Lind, this means that the set of possible values of the entropy of such actions is either all of
2940:
1947:
2702:
404:
3643:
3591:
3544:
2749:
3517:
1338:
1832:
73:
2129:
1886:
1797:
1502:
1373:
2266:
1444:
1303:
382:
349:
252:
221:
201:
2420:
2924:
2660:
2444:
2025:
1906:
1852:
1759:
1665:
1645:
1507:
1464:
1417:
1393:
1237:
272:
4425:
1769:
is sufficient for this, as pointed out by Lind in connection with his study of quasihyperbolic toral automorphisms). As a result, Lehmer was led to ask
1022:
532:
412:
1953:
Some positive answers have been provided as follows, but Lehmer's conjecture is not yet completely proved and is still a question of much interest.
908:
3198:
3073:
3758:
662:
3923:
2185:
3645: or a countable set depending on the solution to Lehmer's problem. Lind also showed that the infinite-dimensional torus either has
4415:
4237:
3983:
3859:
3798:
2449:
1670:
4035:
2757:
2102:
807:
2521:
4337:
3851:
3417:
3338:
3839:
28:
24:
1149:
3750:
3602:
3745:
Smyth, Chris (2008). "The Mahler measure of algebraic numbers: a survey". In McKee, James; Smyth, Chris (eds.).
4288:
2037:
2134:
867:
277:
3835:
2931:
1964:
135:
82:
3999:
Smyth, C. J. (1971). "On the product of the conjugates outside the unit circle of an algebraic integer".
2594:
3057:{\displaystyle {\hat {h}}_{E}(Q)\geq {\frac {C(E/K)}{D}}\left({\frac {\log \log D}{\log D}}\right)^{3},}
2834:
2031:
1396:
3658:
1242:
4420:
4410:
4297:
3681:
46:
1911:
4225:
4221:
3967:
3310:
2669:
902:
387:
4386:
4354:
4313:
3942:
3889:
3719:
3616:
3569:
3522:
2711:
3493:
1762:
1308:
1802:
52:
4383:
4233:
3979:
3855:
3794:
3790:
3754:
3711:
2360:{\displaystyle \log {\mathcal {M}}(P(x))\geq C\left({\frac {\log \log D}{\log D}}\right)^{3}.}
1766:
4346:
4305:
4243:
4202:
4192:
4134:
4092:
4061:
4051:
4016:
4008:
3971:
3932:
3897:
3865:
3804:
3778:
3727:
3701:
3693:
3559:
2515:
2108:
1865:
1776:
1472:
1343:
1273:
3657:, and the Bernoulli shifts are classified up to measurable isomorphism by their entropy by
1761:
does vanish on the circle but not at any root of unity, then the same convergence holds by
1422:
4247:
4206:
4065:
4020:
3869:
3808:
3786:
3731:
3654:
1620:{\displaystyle \Delta _{n}={\text{Res}}(P(x),x^{n}-1)=\prod _{i=1}^{D}(\alpha _{i}^{n}-1)}
1279:
358:
325:
228:
206:
177:
2397:
4301:
2882:
3847:
3831:
3646:
3610:
3321:
Stronger results are known for restricted classes of polynomials or algebraic numbers.
2645:
2429:
2423:
2010:
1891:
1837:
1744:
1650:
1630:
1449:
1402:
1399:
i.e. monic polynomials whose all roots are roots of unity, or a monomial polynomial of
1378:
1222:
1213:
257:
130:
4404:
4358:
3946:
3779:
3774:
2663:
42:
4317:
864:
It is widely believed that this example represents the true minimal value: that is,
16:
Proposed lower bound on the Mahler measure for polynomials with integer coefficients
4176:
3606:
3185:
2518:
function. The canonical height is the analogue for elliptic curves of the function
799:
355:
There are a number of definitions of the Mahler measure, one of which is to factor
1136:{\displaystyle {\mathcal {M}}(P(x))=|a_{0}|\prod _{i=1}^{D}\max(1,|\alpha _{i}|).}
646:{\displaystyle {\mathcal {M}}(P(x))=|a_{0}|\prod _{i=1}^{D}\max(1,|\alpha _{i}|).}
3189:
656:
The smallest known Mahler measure (greater than 1) is for "Lehmer's polynomial"
4123:"On a question of Lehmer and the number of irreducible factors of a polynomial"
2030:
Smyth proved that Lehmer's conjecture is true for all polynomials that are not
516:{\displaystyle P(x)=a_{0}(x-\alpha _{1})(x-\alpha _{2})\cdots (x-\alpha _{D}),}
254:
is an integral multiple of a product of cyclotomic polynomials or the monomial
3975:
3937:
3918:
3706:
3650:
1009:{\displaystyle P(x)=a_{0}(x-\alpha _{1})(x-\alpha _{2})\cdots (x-\alpha _{D})}
76:
3715:
4391:
4139:
4122:
4056:
4039:
3961:
4012:
3902:
3884:
4375:
4284:"Mahler measure and entropy for commuting automorphisms of compact groups"
3299:{\displaystyle {\hat {h}}_{E}(Q)\geq {\frac {C(E/K)}{D^{2}(\log D)^{2}}},}
3174:{\displaystyle {\hat {h}}_{E}(Q)\geq {\frac {C(E/K)}{D^{3}(\log D)^{2}}},}
3067:
due to
Laurent. For arbitrary elliptic curves, the best known result is
4350:
4309:
4197:
4180:
4097:
4080:
3723:
3407:
and this is clearly best possible. If further all the coefficients of
788:{\displaystyle P(x)=x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1\,,}
4332:
4283:
3697:
2105:
and
Stewart independently proved that there is an absolute constant
2250:{\displaystyle \log {\mathcal {M}}(P(x))\geq {\frac {C}{D\log D}}.}
4154:
3609:
of a compact metrizable abelian group is known to be given by the
2374:β₯ 1/1200 and asymptotically C > 1-Ξ΅ for all sufficiently large
4333:"The structure of skew products with ergodic group automorphisms"
4081:"Algebraic integers whose conjugates lie near the unit circle"
27:. For Lehmer's conjecture about Euler's totient function, see
3919:"Dynamical properties of quasihyperbolic toral automorphisms"
1305:
is the logarithm of an algebraic integer. It also shows that
49:. The conjecture asserts that there is an absolute constant
4155:
An effective lower bound for the height of algebraic numbers
2558:
2507:{\displaystyle {\hat {h}}_{E}:E({\bar {K}})\to \mathbb {R} }
2278:
2197:
2140:
1717:
1504:
is an important value in the study of the integer sequences
1248:
1179:
1028:
813:
538:
283:
141:
1734:{\displaystyle \lim |\Delta _{n}|^{1/n}={\mathcal {M}}(P)}
4224:(1990). "On Lehmer's conjecture for elliptic curves". In
3684:(1933). "Factorization of certain cyclotomic functions".
2824:{\displaystyle {\hat {h}}_{E}(Q)\geq {\frac {C(E/K)}{D}}}
854:{\displaystyle {\mathcal {M}}(P(x))=1.176280818\dots \ .}
3963:
Heights of polynomials and entropy in algebraic dynamics
3885:"Speculations concerning the range of Mahler's measure"
3846:. Mathematical Surveys and Monographs. Vol. 104.
3781:
Computational
Excursions in Analysis and Number Theory
2584:{\displaystyle (\deg P)^{-1}\log {\mathcal {M}}(P(x))}
3619:
3572:
3525:
3496:
3420:
3341:
3201:
3076:
2943:
2885:
2837:
2760:
2714:
2672:
2648:
2597:
2524:
2452:
2432:
2400:
2269:
2188:
2137:
2111:
2040:
2013:
1967:
1914:
1894:
1868:
1840:
1805:
1779:
1747:
1673:
1653:
1633:
1510:
1475:
1452:
1425:
1405:
1381:
1346:
1311:
1282:
1245:
1225:
1152:
1025:
911:
870:
810:
665:
535:
415:
390:
361:
328:
280:
260:
231:
209:
180:
138:
85:
55:
19:
For Lehmer's conjecture about the non-vanishing of Ο(
4181:"Counting points of small height on elliptic curves"
3597:
Relation to structure of compact group automorphisms
3476:{\displaystyle M(P)\geq M(x^{2}-x-1)\approx 1.618.}
3397:{\displaystyle M(P)\geq M(x^{3}-x-1)\approx 1.3247}
2934:, then the analogue of Dobrowolski's result holds:
3637:
3585:
3538:
3511:
3475:
3396:
3298:
3173:
3056:
2918:
2867:
2823:
2743:
2696:
2654:
2634:
2583:
2506:
2438:
2414:
2359:
2249:
2171:
2123:
2090:
2019:
1999:
1941:
1900:
1880:
1846:
1826:
1791:
1753:
1733:
1659:
1639:
1619:
1496:
1458:
1438:
1411:
1387:
1367:
1332:
1297:
1264:
1231:
1204:
1135:
1008:
885:
853:
787:
645:
515:
398:
376:
343:
314:
266:
246:
215:
195:
166:
117:
67:
4282:Lind, Douglas; Schmidt, Klaus; Ward, Tom (1990).
3519:be the Mahler measure of the minimal polynomial
1674:
1095:
605:
4232:. Prog. Math. Vol. 91. pp. 103β116.
1205:{\displaystyle m(P)=\log({\mathcal {M}}(P(x))}
2007:be an irreducible monic polynomial of degree
901:Consider Mahler measure for one variable and
8:
4001:Bulletin of the London Mathematical Society
125:satisfies one of the following properties:
1239:has integer coefficients, this shows that
4196:
4138:
4096:
4055:
4040:"Algebraic integers near the unit circle"
3936:
3901:
3705:
3618:
3577:
3571:
3530:
3524:
3495:
3446:
3419:
3367:
3340:
3284:
3262:
3245:
3233:
3215:
3204:
3203:
3200:
3159:
3137:
3120:
3108:
3090:
3079:
3078:
3075:
3045:
3009:
2987:
2975:
2957:
2946:
2945:
2942:
2884:
2851:
2850:
2836:
2804:
2792:
2774:
2763:
2762:
2759:
2724:
2713:
2680:
2679:
2671:
2647:
2611:
2600:
2599:
2596:
2557:
2556:
2541:
2523:
2500:
2499:
2482:
2481:
2466:
2455:
2454:
2451:
2431:
2404:
2399:
2348:
2312:
2277:
2276:
2268:
2223:
2196:
2195:
2187:
2139:
2138:
2136:
2110:
2061:
2045:
2039:
2012:
1984:
1983:
1966:
1913:
1893:
1867:
1839:
1804:
1778:
1746:
1716:
1715:
1702:
1698:
1693:
1686:
1677:
1672:
1652:
1632:
1602:
1597:
1584:
1573:
1551:
1524:
1515:
1509:
1474:
1451:
1430:
1424:
1404:
1380:
1345:
1310:
1281:
1247:
1246:
1244:
1224:
1178:
1177:
1151:
1122:
1116:
1107:
1089:
1078:
1069:
1063:
1054:
1027:
1026:
1024:
997:
972:
950:
931:
910:
869:
812:
811:
809:
781:
763:
750:
737:
724:
711:
698:
685:
664:
632:
626:
617:
599:
588:
579:
573:
564:
537:
536:
534:
501:
476:
454:
435:
414:
392:
391:
389:
360:
327:
282:
281:
279:
259:
230:
208:
179:
140:
139:
137:
102:
101:
84:
54:
4230:SΓ©min. ThΓ©or. Nombres, Paris/Fr. 1988-89
3188:. For elliptic curves with non-integral
4376:http://wayback.cecm.sfu.ca/~mjm/Lehmer/
4268:
4266:
3670:
2091:{\displaystyle x^{D}P(x^{-1})\neq P(x)}
322:. (Equivalently, every complex root of
4378:is a nice reference about the problem.
3960:Everest, Graham; Ward, Thomas (1999).
2172:{\displaystyle {\mathcal {M}}(P(x))=1}
886:{\displaystyle \mu =1.176280818\dots }
315:{\displaystyle {\mathcal {M}}(P(x))=1}
3676:
3674:
3566:, then Lehmer's conjecture holds for
7:
3924:Ergodic Theory and Dynamical Systems
2000:{\displaystyle P(x)\in \mathbb {Z} }
1908:with integer coefficients for which
798:for which the Mahler measure is the
167:{\displaystyle {\mathcal {M}}(P(x))}
118:{\displaystyle P(x)\in \mathbb {Z} }
2635:{\displaystyle {\hat {h}}_{E}(Q)=0}
2034:, i.e., all polynomials satisfying
1667:does not vanish on the circle then
4426:Unsolved problems in number theory
3629:
2868:{\displaystyle Q\in E({\bar {K}})}
1683:
1512:
14:
4387:"Lehmer's Mahler Measure Problem"
2708:asserts that there is a constant
1265:{\displaystyle {\mathcal {M}}(P)}
39:Lehmer's Mahler measure problem,
4157:, Acta Arith. 74 (1996), 81β95.
2370:Dobrowolski obtained the value
3632:
3620:
3506:
3500:
3464:
3439:
3430:
3424:
3385:
3360:
3351:
3345:
3281:
3268:
3253:
3239:
3227:
3221:
3209:
3156:
3143:
3128:
3114:
3102:
3096:
3084:
2995:
2981:
2969:
2963:
2951:
2913:
2904:
2898:
2892:
2862:
2856:
2847:
2812:
2798:
2786:
2780:
2768:
2732:
2718:
2691:
2685:
2676:
2623:
2617:
2605:
2578:
2575:
2569:
2563:
2538:
2525:
2496:
2493:
2487:
2478:
2460:
2298:
2295:
2289:
2283:
2217:
2214:
2208:
2202:
2160:
2157:
2151:
2145:
2085:
2079:
2070:
2054:
1994:
1988:
1977:
1971:
1942:{\displaystyle 0<m(P)<c}
1930:
1924:
1815:
1809:
1765:(in fact an earlier result of
1728:
1722:
1694:
1678:
1614:
1590:
1563:
1541:
1535:
1529:
1485:
1479:
1356:
1350:
1321:
1315:
1292:
1286:
1259:
1253:
1199:
1196:
1190:
1184:
1174:
1162:
1156:
1127:
1123:
1108:
1098:
1070:
1055:
1048:
1045:
1039:
1033:
1003:
984:
978:
959:
956:
937:
921:
915:
833:
830:
824:
818:
675:
669:
637:
633:
618:
608:
580:
565:
558:
555:
549:
543:
507:
488:
482:
463:
460:
441:
425:
419:
371:
365:
338:
332:
303:
300:
294:
288:
241:
235:
190:
184:
161:
158:
152:
146:
112:
106:
95:
89:
1:
4338:Israel Journal of Mathematics
3852:American Mathematical Society
3747:Number Theory and Polynomials
2697:{\displaystyle E({\bar {K}})}
2260:Dobrowolski improved this to
3785:. CMS Books in Mathematics.
3192:, this has been improved to
2426:defined over a number field
2378:. Voutier in 1996 obtained
1773:whether there is a constant
399:{\displaystyle \mathbb {C} }
351:is a root of unity or zero.)
203:is greater than or equal to
3638:{\displaystyle (0,\infty ]}
3586:{\displaystyle P_{\alpha }}
3539:{\displaystyle P_{\alpha }}
2831:for all non-torsion points
2744:{\displaystyle C(E/K)>0}
2591:. It has the property that
4442:
3751:Cambridge University Press
3611:logarithmic Mahler measure
3512:{\displaystyle M(\alpha )}
2706:elliptic Lehmer conjecture
1333:{\displaystyle m(P)\geq 0}
79:with integer coefficients
18:
4416:Theorems in number theory
3976:10.1007/978-1-4471-3898-3
3938:10.1017/s0143385700009573
3603:measure-theoretic entropy
3486:For any algebraic number
3332:) is not reciprocal then
1827:{\displaystyle m(P)>c}
1146:In this paragraph denote
68:{\displaystyle \mu >1}
4289:Inventiones Mathematicae
4121:Dobrowolski, E. (1979).
2926:. If the elliptic curve
893:in Lehmer's conjecture.
29:Lehmer's totient problem
25:Ramanujan's tau function
4140:10.4064/aa-34-4-391-401
4085:Bull. Soc. Math. France
4079:Stewart, C. L. (1978).
4057:10.4064/aa-18-1-355-369
1212:, which is also called
4331:Lind, Douglas (1977).
3903:10.4153/CMB-1981-069-5
3639:
3587:
3540:
3513:
3477:
3398:
3300:
3175:
3058:
2932:complex multiplication
2920:
2869:
2825:
2745:
2698:
2656:
2636:
2585:
2508:
2440:
2416:
2361:
2251:
2173:
2125:
2124:{\displaystyle C>1}
2092:
2021:
2001:
1943:
1902:
1882:
1881:{\displaystyle c>0}
1848:
1828:
1793:
1792:{\displaystyle c>0}
1755:
1735:
1661:
1641:
1621:
1589:
1498:
1497:{\displaystyle m(P)=0}
1460:
1440:
1413:
1397:cyclotomic polynomials
1389:
1369:
1368:{\displaystyle m(P)=0}
1334:
1299:
1266:
1233:
1206:
1137:
1094:
1010:
887:
855:
789:
647:
604:
517:
400:
378:
345:
316:
268:
248:
217:
197:
168:
119:
69:
3838:; Shparlinski, Igor;
3651:measurably isomorphic
3640:
3588:
3541:
3514:
3478:
3399:
3301:
3176:
3059:
2921:
2870:
2826:
2746:
2699:
2657:
2637:
2586:
2509:
2441:
2417:
2362:
2252:
2174:
2126:
2093:
2022:
2002:
1944:
1903:
1883:
1849:
1829:
1794:
1756:
1736:
1662:
1642:
1622:
1569:
1499:
1461:
1441:
1439:{\displaystyle x^{n}}
1414:
1390:
1370:
1335:
1300:
1267:
1234:
1207:
1138:
1074:
1011:
888:
856:
790:
648:
584:
518:
401:
379:
346:
317:
269:
249:
218:
198:
169:
120:
70:
4226:Goldstein, Catherine
4222:Silverman, Joseph H.
4013:10.1112/blms/3.2.169
3917:Lind, D. A. (1982).
3883:Boyd, David (1981).
3844:Recurrence sequences
3836:van der Poorten, Alf
3753:. pp. 322β349.
3617:
3570:
3523:
3494:
3418:
3339:
3199:
3074:
2941:
2883:
2835:
2758:
2712:
2670:
2646:
2595:
2522:
2450:
2430:
2398:
2267:
2186:
2135:
2109:
2038:
2011:
1965:
1912:
1892:
1866:
1838:
1803:
1777:
1745:
1671:
1651:
1631:
1508:
1473:
1469:Lehmer noticed that
1450:
1423:
1403:
1379:
1344:
1309:
1298:{\displaystyle m(P)}
1280:
1243:
1223:
1150:
1023:
909:
868:
808:
663:
533:
413:
388:
377:{\displaystyle P(x)}
359:
344:{\displaystyle P(x)}
326:
278:
258:
247:{\displaystyle P(x)}
229:
216:{\displaystyle \mu }
207:
196:{\displaystyle P(x)}
178:
136:
83:
53:
47:Derrick Henry Lehmer
37:, also known as the
4302:1990InMat.101..593L
4185:Bull. Soc. Math. Fr
2415:{\displaystyle E/K}
1854:is not cyclotomic?,
1607:
35:Lehmer's conjecture
4384:Weisstein, Eric W.
4351:10.1007/BF02759810
4310:10.1007/BF01231517
4272:Smyth (2008) p.329
4260:Smyth (2008) p.328
4198:10.24033/bsmf.2120
4166:Smyth (2008) p.327
4111:Smyth (2008) p.325
4098:10.24033/bsmf.1868
3890:Canad. Math. Bull.
3821:Smyth (2008) p.324
3707:10338.dmlcz/128119
3659:Ornstein's theorem
3635:
3583:
3536:
3509:
3473:
3394:
3317:Restricted results
3296:
3171:
3054:
2919:{\displaystyle D=}
2916:
2865:
2821:
2741:
2694:
2652:
2632:
2581:
2504:
2436:
2412:
2390:Elliptic analogues
2357:
2247:
2169:
2121:
2088:
2017:
1997:
1939:
1898:
1878:
1844:
1824:
1789:
1751:
1731:
1657:
1637:
1617:
1593:
1494:
1456:
1436:
1409:
1385:
1365:
1330:
1295:
1262:
1229:
1202:
1133:
1006:
883:
851:
785:
643:
513:
396:
374:
341:
312:
264:
244:
213:
193:
164:
115:
65:
4036:Montgomery, H. L.
4034:Blanksby, P. E.;
3760:978-0-521-71467-9
3291:
3212:
3166:
3087:
3039:
3002:
2954:
2859:
2819:
2771:
2688:
2655:{\displaystyle Q}
2608:
2490:
2463:
2439:{\displaystyle K}
2342:
2242:
2131:such that either
2020:{\displaystyle D}
1901:{\displaystyle P}
1847:{\displaystyle P}
1754:{\displaystyle P}
1660:{\displaystyle P}
1640:{\displaystyle P}
1527:
1459:{\displaystyle n}
1412:{\displaystyle x}
1388:{\displaystyle P}
1232:{\displaystyle P}
847:
267:{\displaystyle x}
4433:
4397:
4396:
4363:
4362:
4328:
4322:
4321:
4279:
4273:
4270:
4261:
4258:
4252:
4251:
4217:
4211:
4210:
4200:
4173:
4167:
4164:
4158:
4151:
4145:
4144:
4142:
4118:
4112:
4109:
4103:
4102:
4100:
4076:
4070:
4069:
4059:
4031:
4025:
4024:
3996:
3990:
3989:
3957:
3951:
3950:
3940:
3914:
3908:
3907:
3905:
3880:
3874:
3873:
3828:
3822:
3819:
3813:
3812:
3784:
3771:
3765:
3764:
3742:
3736:
3735:
3709:
3678:
3644:
3642:
3641:
3636:
3592:
3590:
3589:
3584:
3582:
3581:
3560:Galois extension
3545:
3543:
3542:
3537:
3535:
3534:
3518:
3516:
3515:
3510:
3482:
3480:
3479:
3474:
3451:
3450:
3403:
3401:
3400:
3395:
3372:
3371:
3305:
3303:
3302:
3297:
3292:
3290:
3289:
3288:
3267:
3266:
3256:
3249:
3234:
3220:
3219:
3214:
3213:
3205:
3180:
3178:
3177:
3172:
3167:
3165:
3164:
3163:
3142:
3141:
3131:
3124:
3109:
3095:
3094:
3089:
3088:
3080:
3063:
3061:
3060:
3055:
3050:
3049:
3044:
3040:
3038:
3027:
3010:
3003:
2998:
2991:
2976:
2962:
2961:
2956:
2955:
2947:
2925:
2923:
2922:
2917:
2874:
2872:
2871:
2866:
2861:
2860:
2852:
2830:
2828:
2827:
2822:
2820:
2815:
2808:
2793:
2779:
2778:
2773:
2772:
2764:
2750:
2748:
2747:
2742:
2728:
2703:
2701:
2700:
2695:
2690:
2689:
2681:
2661:
2659:
2658:
2653:
2641:
2639:
2638:
2633:
2616:
2615:
2610:
2609:
2601:
2590:
2588:
2587:
2582:
2562:
2561:
2549:
2548:
2516:canonical height
2513:
2511:
2510:
2505:
2503:
2492:
2491:
2483:
2471:
2470:
2465:
2464:
2456:
2445:
2443:
2442:
2437:
2421:
2419:
2418:
2413:
2408:
2366:
2364:
2363:
2358:
2353:
2352:
2347:
2343:
2341:
2330:
2313:
2282:
2281:
2256:
2254:
2253:
2248:
2243:
2241:
2224:
2201:
2200:
2178:
2176:
2175:
2170:
2144:
2143:
2130:
2128:
2127:
2122:
2097:
2095:
2094:
2089:
2069:
2068:
2050:
2049:
2026:
2024:
2023:
2018:
2006:
2004:
2003:
1998:
1987:
1948:
1946:
1945:
1940:
1907:
1905:
1904:
1899:
1887:
1885:
1884:
1879:
1853:
1851:
1850:
1845:
1833:
1831:
1830:
1825:
1798:
1796:
1795:
1790:
1760:
1758:
1757:
1752:
1740:
1738:
1737:
1732:
1721:
1720:
1711:
1710:
1706:
1697:
1691:
1690:
1681:
1666:
1664:
1663:
1658:
1646:
1644:
1643:
1638:
1626:
1624:
1623:
1618:
1606:
1601:
1588:
1583:
1556:
1555:
1528:
1525:
1520:
1519:
1503:
1501:
1500:
1495:
1465:
1463:
1462:
1457:
1445:
1443:
1442:
1437:
1435:
1434:
1418:
1416:
1415:
1410:
1395:is a product of
1394:
1392:
1391:
1386:
1374:
1372:
1371:
1366:
1339:
1337:
1336:
1331:
1304:
1302:
1301:
1296:
1274:algebraic number
1271:
1269:
1268:
1263:
1252:
1251:
1238:
1236:
1235:
1230:
1211:
1209:
1208:
1203:
1183:
1182:
1142:
1140:
1139:
1134:
1126:
1121:
1120:
1111:
1093:
1088:
1073:
1068:
1067:
1058:
1032:
1031:
1015:
1013:
1012:
1007:
1002:
1001:
977:
976:
955:
954:
936:
935:
903:Jensen's formula
892:
890:
889:
884:
860:
858:
857:
852:
845:
817:
816:
794:
792:
791:
786:
768:
767:
755:
754:
742:
741:
729:
728:
716:
715:
703:
702:
690:
689:
652:
650:
649:
644:
636:
631:
630:
621:
603:
598:
583:
578:
577:
568:
542:
541:
522:
520:
519:
514:
506:
505:
481:
480:
459:
458:
440:
439:
405:
403:
402:
397:
395:
383:
381:
380:
375:
350:
348:
347:
342:
321:
319:
318:
313:
287:
286:
274:, in which case
273:
271:
270:
265:
253:
251:
250:
245:
222:
220:
219:
214:
202:
200:
199:
194:
173:
171:
170:
165:
145:
144:
124:
122:
121:
116:
105:
75:such that every
74:
72:
71:
66:
41:is a problem in
4441:
4440:
4436:
4435:
4434:
4432:
4431:
4430:
4401:
4400:
4382:
4381:
4372:
4367:
4366:
4330:
4329:
4325:
4281:
4280:
4276:
4271:
4264:
4259:
4255:
4240:
4219:
4218:
4214:
4175:
4174:
4170:
4165:
4161:
4152:
4148:
4120:
4119:
4115:
4110:
4106:
4078:
4077:
4073:
4033:
4032:
4028:
3998:
3997:
3993:
3986:
3959:
3958:
3954:
3916:
3915:
3911:
3882:
3881:
3877:
3862:
3832:Everest, Graham
3830:
3829:
3825:
3820:
3816:
3801:
3787:Springer-Verlag
3773:
3772:
3768:
3761:
3744:
3743:
3739:
3698:10.2307/1968172
3680:
3679:
3672:
3667:
3655:Bernoulli shift
3615:
3614:
3599:
3573:
3568:
3567:
3550:. If the field
3526:
3521:
3520:
3492:
3491:
3442:
3416:
3415:
3363:
3337:
3336:
3319:
3280:
3258:
3257:
3235:
3202:
3197:
3196:
3155:
3133:
3132:
3110:
3077:
3072:
3071:
3028:
3011:
3005:
3004:
2977:
2944:
2939:
2938:
2881:
2880:
2833:
2832:
2794:
2761:
2756:
2755:
2710:
2709:
2668:
2667:
2644:
2643:
2642:if and only if
2598:
2593:
2592:
2537:
2520:
2519:
2453:
2448:
2447:
2428:
2427:
2396:
2395:
2392:
2331:
2314:
2308:
2307:
2265:
2264:
2228:
2184:
2183:
2133:
2132:
2107:
2106:
2057:
2041:
2036:
2035:
2009:
2008:
1963:
1962:
1959:
1957:Partial results
1910:
1909:
1890:
1889:
1864:
1863:
1836:
1835:
1801:
1800:
1775:
1774:
1763:Baker's theorem
1743:
1742:
1692:
1682:
1669:
1668:
1649:
1648:
1629:
1628:
1547:
1511:
1506:
1505:
1471:
1470:
1448:
1447:
1426:
1421:
1420:
1401:
1400:
1377:
1376:
1342:
1341:
1307:
1306:
1278:
1277:
1241:
1240:
1221:
1220:
1148:
1147:
1112:
1059:
1021:
1020:
993:
968:
946:
927:
907:
906:
899:
866:
865:
806:
805:
759:
746:
733:
720:
707:
694:
681:
661:
660:
622:
569:
531:
530:
497:
472:
450:
431:
411:
410:
386:
385:
357:
356:
324:
323:
276:
275:
256:
255:
227:
226:
205:
204:
176:
175:
134:
133:
81:
80:
51:
50:
32:
17:
12:
11:
5:
4439:
4437:
4429:
4428:
4423:
4418:
4413:
4403:
4402:
4399:
4398:
4379:
4371:
4370:External links
4368:
4365:
4364:
4345:(3): 205β248.
4323:
4274:
4262:
4253:
4238:
4220:Hindry, Marc;
4212:
4191:(2): 247β265.
4168:
4159:
4146:
4133:(4): 391β401.
4113:
4104:
4071:
4026:
4007:(2): 169β175.
3991:
3984:
3952:
3909:
3896:(4): 453β469.
3875:
3860:
3854:. p. 30.
3848:Providence, RI
3823:
3814:
3799:
3775:Borwein, Peter
3766:
3759:
3737:
3692:(3): 461β479.
3669:
3668:
3666:
3663:
3634:
3631:
3628:
3625:
3622:
3605:of an ergodic
3598:
3595:
3580:
3576:
3533:
3529:
3508:
3505:
3502:
3499:
3484:
3483:
3472:
3469:
3466:
3463:
3460:
3457:
3454:
3449:
3445:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3405:
3404:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3370:
3366:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3318:
3315:
3309:by Hindry and
3307:
3306:
3295:
3287:
3283:
3279:
3276:
3273:
3270:
3265:
3261:
3255:
3252:
3248:
3244:
3241:
3238:
3232:
3229:
3226:
3223:
3218:
3211:
3208:
3182:
3181:
3170:
3162:
3158:
3154:
3151:
3148:
3145:
3140:
3136:
3130:
3127:
3123:
3119:
3116:
3113:
3107:
3104:
3101:
3098:
3093:
3086:
3083:
3065:
3064:
3053:
3048:
3043:
3037:
3034:
3031:
3026:
3023:
3020:
3017:
3014:
3008:
3001:
2997:
2994:
2990:
2986:
2983:
2980:
2974:
2971:
2968:
2965:
2960:
2953:
2950:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2877:
2876:
2864:
2858:
2855:
2849:
2846:
2843:
2840:
2818:
2814:
2811:
2807:
2803:
2800:
2797:
2791:
2788:
2785:
2782:
2777:
2770:
2767:
2740:
2737:
2734:
2731:
2727:
2723:
2720:
2717:
2693:
2687:
2684:
2678:
2675:
2651:
2631:
2628:
2625:
2622:
2619:
2614:
2607:
2604:
2580:
2577:
2574:
2571:
2568:
2565:
2560:
2555:
2552:
2547:
2544:
2540:
2536:
2533:
2530:
2527:
2502:
2498:
2495:
2489:
2486:
2480:
2477:
2474:
2469:
2462:
2459:
2435:
2424:elliptic curve
2411:
2407:
2403:
2391:
2388:
2368:
2367:
2356:
2351:
2346:
2340:
2337:
2334:
2329:
2326:
2323:
2320:
2317:
2311:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2280:
2275:
2272:
2258:
2257:
2246:
2240:
2237:
2234:
2231:
2227:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2199:
2194:
2191:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2142:
2120:
2117:
2114:
2087:
2084:
2081:
2078:
2075:
2072:
2067:
2064:
2060:
2056:
2053:
2048:
2044:
2016:
1996:
1993:
1990:
1986:
1982:
1979:
1976:
1973:
1970:
1958:
1955:
1951:
1950:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1897:
1877:
1874:
1871:
1856:
1855:
1843:
1823:
1820:
1817:
1814:
1811:
1808:
1788:
1785:
1782:
1750:
1730:
1727:
1724:
1719:
1714:
1709:
1705:
1701:
1696:
1689:
1685:
1680:
1676:
1656:
1636:
1616:
1613:
1610:
1605:
1600:
1596:
1592:
1587:
1582:
1579:
1576:
1572:
1568:
1565:
1562:
1559:
1554:
1550:
1546:
1543:
1540:
1537:
1534:
1531:
1523:
1518:
1514:
1493:
1490:
1487:
1484:
1481:
1478:
1455:
1433:
1429:
1408:
1384:
1364:
1361:
1358:
1355:
1352:
1349:
1329:
1326:
1323:
1320:
1317:
1314:
1294:
1291:
1288:
1285:
1261:
1258:
1255:
1250:
1228:
1214:Mahler measure
1201:
1198:
1195:
1192:
1189:
1186:
1181:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1144:
1143:
1132:
1129:
1125:
1119:
1115:
1110:
1106:
1103:
1100:
1097:
1092:
1087:
1084:
1081:
1077:
1072:
1066:
1062:
1057:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1030:
1005:
1000:
996:
992:
989:
986:
983:
980:
975:
971:
967:
964:
961:
958:
953:
949:
945:
942:
939:
934:
930:
926:
923:
920:
917:
914:
905:shows that if
898:
895:
882:
879:
876:
873:
862:
861:
850:
844:
841:
838:
835:
832:
829:
826:
823:
820:
815:
796:
795:
784:
780:
777:
774:
771:
766:
762:
758:
753:
749:
745:
740:
736:
732:
727:
723:
719:
714:
710:
706:
701:
697:
693:
688:
684:
680:
677:
674:
671:
668:
654:
653:
642:
639:
635:
629:
625:
620:
616:
613:
610:
607:
602:
597:
594:
591:
587:
582:
576:
572:
567:
563:
560:
557:
554:
551:
548:
545:
540:
524:
523:
512:
509:
504:
500:
496:
493:
490:
487:
484:
479:
475:
471:
468:
465:
462:
457:
453:
449:
446:
443:
438:
434:
430:
427:
424:
421:
418:
394:
373:
370:
367:
364:
353:
352:
340:
337:
334:
331:
311:
308:
305:
302:
299:
296:
293:
290:
285:
263:
243:
240:
237:
234:
224:
212:
192:
189:
186:
183:
163:
160:
157:
154:
151:
148:
143:
131:Mahler measure
114:
111:
108:
104:
100:
97:
94:
91:
88:
64:
61:
58:
15:
13:
10:
9:
6:
4:
3:
2:
4438:
4427:
4424:
4422:
4419:
4417:
4414:
4412:
4409:
4408:
4406:
4394:
4393:
4388:
4385:
4380:
4377:
4374:
4373:
4369:
4360:
4356:
4352:
4348:
4344:
4340:
4339:
4334:
4327:
4324:
4319:
4315:
4311:
4307:
4303:
4299:
4295:
4291:
4290:
4285:
4278:
4275:
4269:
4267:
4263:
4257:
4254:
4249:
4245:
4241:
4239:0-8176-3493-2
4235:
4231:
4227:
4223:
4216:
4213:
4208:
4204:
4199:
4194:
4190:
4186:
4182:
4178:
4172:
4169:
4163:
4160:
4156:
4150:
4147:
4141:
4136:
4132:
4128:
4124:
4117:
4114:
4108:
4105:
4099:
4094:
4090:
4086:
4082:
4075:
4072:
4067:
4063:
4058:
4053:
4049:
4045:
4041:
4037:
4030:
4027:
4022:
4018:
4014:
4010:
4006:
4002:
3995:
3992:
3987:
3985:1-85233-125-9
3981:
3977:
3973:
3969:
3965:
3964:
3956:
3953:
3948:
3944:
3939:
3934:
3930:
3926:
3925:
3920:
3913:
3910:
3904:
3899:
3895:
3892:
3891:
3886:
3879:
3876:
3871:
3867:
3863:
3861:0-8218-3387-1
3857:
3853:
3849:
3845:
3841:
3837:
3833:
3827:
3824:
3818:
3815:
3810:
3806:
3802:
3800:0-387-95444-9
3796:
3792:
3788:
3783:
3782:
3776:
3770:
3767:
3762:
3756:
3752:
3748:
3741:
3738:
3733:
3729:
3725:
3721:
3717:
3713:
3708:
3703:
3699:
3695:
3691:
3687:
3683:
3677:
3675:
3671:
3664:
3662:
3660:
3656:
3652:
3648:
3626:
3623:
3612:
3608:
3604:
3596:
3594:
3578:
3574:
3565:
3561:
3557:
3553:
3549:
3531:
3527:
3503:
3497:
3489:
3470:
3467:
3461:
3458:
3455:
3452:
3447:
3443:
3436:
3433:
3427:
3421:
3414:
3413:
3412:
3411:are odd then
3410:
3391:
3388:
3382:
3379:
3376:
3373:
3368:
3364:
3357:
3354:
3348:
3342:
3335:
3334:
3333:
3331:
3327:
3322:
3316:
3314:
3312:
3293:
3285:
3277:
3274:
3271:
3263:
3259:
3250:
3246:
3242:
3236:
3230:
3224:
3216:
3206:
3195:
3194:
3193:
3191:
3187:
3168:
3160:
3152:
3149:
3146:
3138:
3134:
3125:
3121:
3117:
3111:
3105:
3099:
3091:
3081:
3070:
3069:
3068:
3051:
3046:
3041:
3035:
3032:
3029:
3024:
3021:
3018:
3015:
3012:
3006:
2999:
2992:
2988:
2984:
2978:
2972:
2966:
2958:
2948:
2937:
2936:
2935:
2933:
2929:
2910:
2907:
2901:
2895:
2889:
2886:
2853:
2844:
2841:
2838:
2816:
2809:
2805:
2801:
2795:
2789:
2783:
2775:
2765:
2754:
2753:
2752:
2738:
2735:
2729:
2725:
2721:
2715:
2707:
2682:
2673:
2665:
2664:torsion point
2649:
2629:
2626:
2620:
2612:
2602:
2572:
2566:
2553:
2550:
2545:
2542:
2534:
2531:
2528:
2517:
2484:
2475:
2472:
2467:
2457:
2433:
2425:
2409:
2405:
2401:
2389:
2387:
2385:
2381:
2377:
2373:
2354:
2349:
2344:
2338:
2335:
2332:
2327:
2324:
2321:
2318:
2315:
2309:
2304:
2301:
2292:
2286:
2273:
2270:
2263:
2262:
2261:
2244:
2238:
2235:
2232:
2229:
2225:
2220:
2211:
2205:
2192:
2189:
2182:
2181:
2180:
2166:
2163:
2154:
2148:
2118:
2115:
2112:
2104:
2101:Blanksby and
2099:
2082:
2076:
2073:
2065:
2062:
2058:
2051:
2046:
2042:
2033:
2028:
2014:
1991:
1980:
1974:
1968:
1956:
1954:
1936:
1933:
1927:
1921:
1918:
1915:
1895:
1875:
1872:
1869:
1861:
1860:
1859:
1841:
1821:
1818:
1812:
1806:
1786:
1783:
1780:
1772:
1771:
1770:
1768:
1764:
1748:
1725:
1712:
1707:
1703:
1699:
1687:
1654:
1634:
1611:
1608:
1603:
1598:
1594:
1585:
1580:
1577:
1574:
1570:
1566:
1560:
1557:
1552:
1548:
1544:
1538:
1532:
1521:
1516:
1491:
1488:
1482:
1476:
1467:
1453:
1431:
1427:
1419:i.e. a power
1406:
1398:
1382:
1362:
1359:
1353:
1347:
1327:
1324:
1318:
1312:
1289:
1283:
1275:
1256:
1226:
1217:
1215:
1193:
1187:
1171:
1168:
1165:
1159:
1153:
1130:
1117:
1113:
1104:
1101:
1090:
1085:
1082:
1079:
1075:
1064:
1060:
1051:
1042:
1036:
1019:
1018:
1017:
998:
994:
990:
987:
981:
973:
969:
965:
962:
951:
947:
943:
940:
932:
928:
924:
918:
912:
904:
896:
894:
880:
877:
874:
871:
848:
842:
839:
836:
827:
821:
804:
803:
802:
801:
782:
778:
775:
772:
769:
764:
760:
756:
751:
747:
743:
738:
734:
730:
725:
721:
717:
712:
708:
704:
699:
695:
691:
686:
682:
678:
672:
666:
659:
658:
657:
640:
627:
623:
614:
611:
600:
595:
592:
589:
585:
574:
570:
561:
552:
546:
529:
528:
527:
526:and then set
510:
502:
498:
494:
491:
485:
477:
473:
469:
466:
455:
451:
447:
444:
436:
432:
428:
422:
416:
409:
408:
407:
368:
362:
335:
329:
309:
306:
297:
291:
261:
238:
232:
225:
210:
187:
181:
155:
149:
132:
128:
127:
126:
109:
98:
92:
86:
78:
62:
59:
56:
48:
44:
43:number theory
40:
36:
30:
26:
22:
4390:
4342:
4336:
4326:
4293:
4287:
4277:
4256:
4229:
4215:
4188:
4184:
4177:Masser, D.W.
4171:
4162:
4153:P. Voutier,
4149:
4130:
4126:
4116:
4107:
4088:
4084:
4074:
4047:
4043:
4029:
4004:
4000:
3994:
3962:
3955:
3928:
3922:
3912:
3893:
3888:
3878:
3843:
3840:Ward, Thomas
3826:
3817:
3780:
3769:
3746:
3740:
3689:
3685:
3682:Lehmer, D.H.
3607:automorphism
3600:
3563:
3555:
3551:
3547:
3487:
3485:
3408:
3406:
3329:
3325:
3323:
3320:
3308:
3183:
3066:
2927:
2878:
2705:
2393:
2383:
2379:
2375:
2371:
2369:
2259:
2100:
2029:
1960:
1952:
1888:, are there
1857:
1468:
1340:and that if
1218:
1145:
900:
863:
800:Salem number
797:
655:
525:
354:
38:
34:
33:
20:
4421:Conjectures
4411:Polynomials
4296:: 593β629.
4091:: 169β176.
4050:: 355β369.
3190:j-invariant
878:1.176280818
840:1.176280818
4405:Categories
4248:0741.14013
4207:0723.14026
4127:Acta Arith
4066:0221.12003
4044:Acta Arith
4021:1139.11002
3966:. London:
3870:1033.11006
3809:1020.12001
3789:. p.
3732:0007.19904
3665:References
2751:such that
2446:, and let
2382:β₯ 1/4 for
2103:Montgomery
2032:reciprocal
1799:such that
1627:for monic
897:Motivation
77:polynomial
45:raised by
4392:MathWorld
4359:120160631
3947:120859454
3931:: 49β68.
3716:0003-486X
3686:Ann. Math
3630:∞
3579:α
3532:α
3504:α
3468:≈
3459:−
3453:−
3434:≥
3389:≈
3380:−
3374:−
3355:≥
3311:Silverman
3275:
3231:≥
3210:^
3150:
3106:≥
3085:^
3033:
3022:
3016:
2973:≥
2952:^
2857:¯
2842:∈
2790:≥
2769:^
2686:¯
2606:^
2554:
2543:−
2532:
2497:→
2488:¯
2461:^
2336:
2325:
2319:
2302:≥
2274:
2236:
2221:≥
2193:
2074:≠
2063:−
1981:∈
1834:provided
1684:Δ
1609:−
1595:α
1571:∏
1558:−
1513:Δ
1446:for some
1325:≥
1172:
1114:α
1076:∏
995:α
991:−
982:⋯
970:α
966:−
948:α
944:−
881:…
872:μ
843:…
757:−
744:−
731:−
718:−
705:−
624:α
586:∏
499:α
495:−
486:⋯
474:α
470:−
452:α
448:−
211:μ
99:∈
57:μ
4318:17077751
4179:(1989).
4038:(1971).
3968:Springer
3842:(2003).
3777:(2002).
4298:Bibcode
4228:(ed.).
3724:1968172
3647:ergodic
3558:) is a
3184:due to
2514:be the
1767:Gelfond
1016:then
23:), see
4357:
4316:
4246:
4236:
4205:
4064:
4019:
3982:
3945:
3868:
3858:
3807:
3797:
3757:
3730:
3722:
3714:
3490:, let
3471:1.618.
3392:1.3247
3186:Masser
2879:where
2704:. The
2422:be an
1862:given
1741:. If
1272:is an
846:
4355:S2CID
4314:S2CID
3943:S2CID
3720:JSTOR
3688:. 2.
3653:to a
2662:is a
2386:β₯ 2.
1647:. If
1375:then
384:over
4234:ISBN
3980:ISBN
3856:ISBN
3795:ISBN
3755:ISBN
3712:ISSN
3601:The
2930:has
2736:>
2394:Let
2116:>
1961:Let
1934:<
1919:<
1873:>
1819:>
1784:>
129:The
60:>
4347:doi
4306:doi
4294:101
4244:Zbl
4203:Zbl
4193:doi
4189:117
4135:doi
4093:doi
4089:106
4062:Zbl
4052:doi
4017:Zbl
4009:doi
3972:doi
3933:doi
3898:doi
3866:Zbl
3805:Zbl
3728:Zbl
3702:hdl
3694:doi
3562:of
3546:of
3324:If
3272:log
3147:log
3030:log
3019:log
3013:log
2666:in
2551:log
2529:deg
2333:log
2322:log
2316:log
2271:log
2233:log
2190:log
2179:or
1858:or
1675:lim
1526:Res
1276:so
1219:If
1169:log
1096:max
606:max
406:as
174:of
4407::
4389:.
4353:.
4343:28
4341:.
4335:.
4312:.
4304:.
4292:.
4286:.
4265:^
4242:.
4201:.
4187:.
4183:.
4131:34
4129:.
4125:.
4087:.
4083:.
4060:.
4048:18
4046:.
4042:.
4015:.
4003:.
3978:.
3970:.
3941:.
3927:.
3921:.
3894:24
3887:.
3864:.
3850::
3834:;
3803:.
3793:.
3791:16
3749:.
3726:.
3718:.
3710:.
3700:.
3690:34
3673:^
3593:.
3313:.
2098:.
2027:.
1466:.
1216:.
687:10
4395:.
4361:.
4349::
4320:.
4308::
4300::
4250:.
4209:.
4195::
4143:.
4137::
4101:.
4095::
4068:.
4054::
4023:.
4011::
4005:3
3988:.
3974::
3949:.
3935::
3929:2
3906:.
3900::
3872:.
3811:.
3763:.
3734:.
3704::
3696::
3633:]
3627:,
3624:0
3621:(
3575:P
3564:Q
3556:Ξ±
3554:(
3552:Q
3548:Ξ±
3528:P
3507:)
3501:(
3498:M
3488:Ξ±
3465:)
3462:1
3456:x
3448:2
3444:x
3440:(
3437:M
3431:)
3428:P
3425:(
3422:M
3409:P
3386:)
3383:1
3377:x
3369:3
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3294:,
3286:2
3282:)
3278:D
3269:(
3264:2
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3254:)
3251:K
3247:/
3243:E
3240:(
3237:C
3228:)
3225:Q
3222:(
3217:E
3207:h
3169:,
3161:2
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3144:(
3139:3
3135:D
3129:)
3126:K
3122:/
3118:E
3115:(
3112:C
3103:)
3100:Q
3097:(
3092:E
3082:h
3052:,
3047:3
3042:)
3036:D
3025:D
3007:(
3000:D
2996:)
2993:K
2989:/
2985:E
2982:(
2979:C
2970:)
2967:Q
2964:(
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2949:h
2928:E
2914:]
2911:K
2908::
2905:)
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2899:(
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2893:[
2890:=
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2875:,
2863:)
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2806:/
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2799:(
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2726:/
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2624:)
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713:7
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90:(
87:P
63:1
31:.
21:n
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