3681:
319:
2964:
1838:
1862:
43:
1562:
1413:
2899:
887:
2578:
1438:
1256:
2745:
1200:
718:
1808:
302:
502:
2469:
2687:
2139:
733:
1732:
1083:
954:
1557:{\displaystyle \kappa =\left\lfloor a+{\tfrac {1}{\sqrt {3}}}b\right\rceil +\left\lfloor {\tfrac {2}{\sqrt {3}}}b\right\rceil \omega \ \ {\text{ and }}\ \ \rho ={\alpha }-\kappa \beta .}
2729:
565:
3781:
3246:
3432:
3385:
3204:
1591:
1408:{\displaystyle {\frac {\alpha }{\beta }}\ =\ {\tfrac {1}{\ |\beta |^{2}}}\alpha {\overline {\beta }}\ =\ a+bi\ =\ a+{\tfrac {1}{\sqrt {3}}}b+{\tfrac {2}{\sqrt {3}}}b\omega ,}
3812:
3343:
2072:
202:
3717:
2619:
3166:
2039:
1123:
2894:{\displaystyle \sum _{z\in \mathbf {E} \setminus \{0\}}{\frac {1}{z^{6}}}=G_{6}\left(e^{\frac {2\pi i}{3}}\right)={\frac {\Gamma (1/3)^{18}}{8960\pi ^{6}}}}
3501:
596:
2218:
28:
3710:
3848:
1737:
3128:
3302:
2960:
among all such complex tori. This torus can be obtained by identifying each of the three pairs of opposite edges of a regular hexagon.
60:
229:
1244:
One division algorithm is as follows. First perform the division in the field of complex numbers, and write the quotient in terms of
403:
3703:
3112:
1238:
126:
107:
3575:
3494:
2628:
79:
64:
3740:
3013:
882:{\displaystyle {\left|a+b\;\!\omega \right|}^{2}\,=\,{(a-{\tfrac {1}{2}}b)}^{2}+{\tfrac {3}{4}}b^{2}\,=\,a^{2}-ab+b^{2}~,}
86:
2081:
3288:
3589:
3063:
call these numbers "Euler-egészek", that is, Eulerian integers. The latter claims Euler worked with them in a proof.
2975:, and can be obtained by identifying each of the two pairs of opposite sides of a square fundamental domain, such as
2007:, and these factors are Eisenstein primes: they are precisely the Eisenstein integers whose norm is a rational prime.
1693:
1011:
2573:{\displaystyle \sum _{z\in \mathbf {E} \setminus \{0\}}{\frac {1}{z^{4}}}=G_{4}\left(e^{\frac {2\pi i}{3}}\right)=0}
911:
93:
3863:
3853:
3843:
3487:
2420:
53:
3028:
2692:
519:
75:
3858:
3750:
2075:
1432:. Then obtain the Eisenstein integer quotient by rounding the rational coefficients to the nearest integer:
3209:
363:
3536:
3390:
3348:
3680:
3171:
3099:
1570:
3760:
3531:
3306:
2971:
The other maximally symmetric torus is the quotient of the complex plane by the additive lattice of
2044:
1230:
896:
169:
148:
2583:
3786:
3685:
3018:
1089:
327:
312:
3745:
100:
3132:
2018:
3868:
3822:
3726:
3666:
3657:
3648:
3546:
3521:
3108:
3074:
3023:
3008:
2998:
2972:
2917:
1841:
Small
Eisenstein primes. Those on the green axes are associate to a natural prime of the form
1620:
1234:
359:
335:
1849:. All others have an absolute value equal to 3 or square root of a natural prime of the form
3817:
3796:
3643:
3638:
3633:
3628:
3623:
3556:
3541:
3510:
3003:
2993:
2988:
1831:
996:
395:
377:
355:
318:
3594:
3526:
2963:
2945:
1195:{\displaystyle \alpha =\kappa \beta +\rho \ \ {\text{ with }}\ \ N(\rho )<N(\beta ).}
724:
3791:
2739:
2431:
2409:
1928:
968:
960:
339:
160:
156:
3474:
322:
Eisenstein integers as the points of a certain triangular lattice in the complex plane
3837:
3755:
3599:
3580:
3570:
3077:
2949:
2935:
1646:, acting by translations on the complex plane, is the 60°–120° rhombus with vertices
343:
331:
308:
713:{\displaystyle (a+b\;\!\omega )\;\!(c+d\;\!\omega )=(ac-bd)+(bc+ad-bd)\;\!\omega ~.}
3609:
3604:
3561:
2208:
1939:
1837:
964:
17:
3267:
3101:
Primes of the Form x2+ny2: Fermat, Class Field Theory and
Complex Multiplication
2622:
2204:
2200:
2196:
2192:
2188:
2184:
2180:
2176:
2172:
2168:
2164:
1915:
is said to be an
Eisenstein prime if its only non-unit divisors are of the form
1861:
140:
42:
34:
Complex number whose mapping on a coordinate plane produces a triangular lattice
2160:
2156:
3261:
3082:
2427:
380:. To see that the Eisenstein integers are algebraic integers note that each
2967:
Identifying each of the three pairs of opposite edges of a regular hexagon.
2400:
Up to conjugacy and unit multiples, the primes listed above, together with
1690:
to this vertex, but the maximum possible distance in our algorithm is only
1678:
lies inside one of the translates of this parallelogram, and the quotient
27:"Eulerian integer" and "Euler integer" redirect here. For other uses, see
2957:
1830:
For the unrelated concept of an
Eisenstein prime of a modular curve, see
1594:
2419:
As of
October 2023, the largest known real Eisenstein prime is the
1803:{\displaystyle |\rho |\leq {\tfrac {\sqrt {3}}{2}}|\beta |<|\beta |}
220:
3695:
3479:
3447:
1927:
is any of the six units. They are the corresponding concept to the
2962:
2435:
1860:
1836:
1684:
is one of its vertices. The remainder is the square distance from
297:{\displaystyle \omega ={\frac {-1+i{\sqrt {3}}}{2}}=e^{i2\pi /3}}
2734:
The sum of the reciprocals of all
Eisenstein integers excluding
2458:
The sum of the reciprocals of all
Eisenstein integers excluding
29:
List of topics named after
Leonhard Euler § Euler's numbers
3699:
3483:
497:{\displaystyle z^{2}-(2a-b)\;\!z+\left(a^{2}-ab+b^{2}\right)~.}
36:
3448:"Entry 0fda1b – Fungrim: The Mathematical Functions Grimoire"
2738:
raised to the sixth power can be expressed in terms of the
2241:
Eisenstein primes: they admit nontrivial factorizations in
2213:
2682:{\displaystyle G_{k}\left(e^{\frac {2\pi i}{3}}\right)=0}
1623:
rings, is as follows. A fundamental domain for the ideal
892:
which is clearly a positive ordinary (rational) integer.
1229:
are all
Eisenstein integers. This algorithm implies the
1758:
1698:
1490:
1460:
1383:
1363:
1280:
815:
787:
3393:
3351:
3309:
3212:
3174:
3135:
2748:
2695:
2631:
2586:
2472:
2141:, so it is regarded as a special type in some books.
2084:
2047:
2021:
1740:
1696:
1619:, while the analogous procedure fails for most other
1573:
1441:
1259:
1126:
1014:
914:
736:
599:
522:
406:
232:
172:
3782:
Gromov's systolic inequality for essential manifolds
342:
in the complex plane. The
Eisenstein integers are a
3805:
3774:
3733:
2134:{\displaystyle 1-\omega =(-\omega )(1-\omega ^{2})}
67:. Unsourced material may be challenged and removed.
3426:
3379:
3337:
3240:
3198:
3160:
2893:
2723:
2681:
2613:
2572:
2430:. With one exception, all larger known primes are
2133:
2066:
2033:
1802:
1726:
1585:
1556:
1407:
1194:
1077:
948:
881:
712:
559:
496:
296:
196:
2450:; thus no Mersenne prime is an Eisenstein prime.
754:
700:
635:
621:
613:
439:
3813:Gromov's inequality for complex projective space
723:The 2-norm of an Eisenstein integer is just its
1727:{\displaystyle {\tfrac {\sqrt {3}}{2}}|\beta |}
1241:of Eisenstein integers into Eisenstein primes.
1078:{\displaystyle N(a+b\,\omega )=a^{2}-ab+b^{2}.}
949:{\displaystyle {\bar {\omega }}=\omega ^{2}~.}
3711:
3495:
8:
2774:
2768:
2498:
2492:
2144:The first few Eisenstein primes of the form
1580:
1574:
3718:
3704:
3696:
3502:
3488:
3480:
2438:. Real Eisenstein primes are congruent to
1005:is given by the square modulus, as above:
753:
699:
634:
620:
612:
438:
3414:
3404:
3392:
3356:
3350:
3314:
3308:
3222:
3211:
3181:
3177:
3176:
3173:
3140:
3134:
2882:
2867:
2855:
2843:
2817:
2803:
2788:
2779:
2760:
2753:
2747:
2724:{\displaystyle k\not \equiv 0{\pmod {6}}}
2705:
2694:
2650:
2636:
2630:
2601:
2591:
2585:
2541:
2527:
2512:
2503:
2484:
2477:
2471:
2122:
2083:
2058:
2046:
2020:
1934:There are two types of Eisenstein prime.
1795:
1787:
1779:
1771:
1757:
1749:
1741:
1739:
1719:
1711:
1697:
1695:
1572:
1537:
1520:
1489:
1459:
1440:
1382:
1362:
1316:
1303:
1298:
1289:
1279:
1260:
1258:
1151:
1125:
1066:
1044:
1030:
1013:
934:
916:
915:
913:
867:
845:
840:
836:
830:
814:
805:
786:
776:
774:
770:
764:
738:
735:
598:
560:{\displaystyle \omega ^{2}+\omega +1=0~.}
527:
521:
477:
455:
411:
405:
284:
274:
254:
239:
231:
171:
127:Learn how and when to remove this message
2948:containing all Eisenstein integers is a
1984:. Thus, such a prime may be factored as
995:The ring of Eisenstein integers forms a
317:
3289:"What are the zeros of the j-function?"
3040:
2956:. This is one of two tori with maximal
2765:
2489:
2442:, and all Mersenne primes greater than
983:, the Eisenstein integers of norm
570:The product of two Eisenstein integers
1816:could be slightly decreased by taking
1117:smaller than the divisor, satisfying:
2225:Natural primes that are congruent to
1956:and each rational prime congruent to
1881:are Eisenstein integers, we say that
7:
3241:{\displaystyle p\equiv 1{\pmod {3}}}
2344:Some non-real Eisenstein primes are
1893:if there is some Eisenstein integer
65:adding citations to reliable sources
3230:
2713:
2426:, discovered by Péter Szabolcs and
2408:, are all the Eisenstein primes of
1865:Eisenstein primes in a larger range
3475:Eisenstein Integer--from MathWorld
3427:{\displaystyle \rho =e^{2\pi i/3}}
3380:{\displaystyle G_{6}(\rho )\neq 0}
2846:
25:
3679:
3199:{\displaystyle \mathbb {F} _{p}}
2907:are the Eisenstein integers and
2761:
2485:
2289:and can therefore be written as
1909:. A non-unit Eisenstein integer
1586:{\displaystyle \lfloor x\rceil }
41:
3223:
2706:
2275:In general, if a natural prime
2011:In the second type, factors of
1593:may denote any of the standard
354:The Eisenstein integers form a
326:The Eisenstein integers form a
52:needs additional citations for
3368:
3362:
3338:{\displaystyle G_{4}(i)\neq 0}
3326:
3320:
3234:
3224:
3193:
3187:
2864:
2849:
2717:
2707:
2462:raised to the fourth power is
2128:
2109:
2106:
2097:
1796:
1788:
1780:
1772:
1750:
1742:
1720:
1712:
1299:
1290:
1186:
1180:
1171:
1165:
1034:
1018:
921:
801:
777:
696:
669:
663:
645:
639:
622:
617:
600:
435:
420:
151:), occasionally also known as
1:
2067:{\displaystyle 1-\omega ^{2}}
197:{\displaystyle z=a+b\omega ,}
3849:Quadratic irrational numbers
3098:Cox, David A. (1997-05-08).
3061:. Tankönyvkiadó. p. 75.
2614:{\displaystyle e^{2\pi i/3}}
1950:is also an Eisenstein prime.
1321:
3590:Quadratic irrational number
3576:Pisot–Vijayaraghavan number
1822:to be the closest corner.)
3885:
3741:Loewner's torus inequality
3014:Loewner's torus inequality
2930:by the Eisenstein integers
2307:, then it factorizes over
1931:in the Gaussian integers.
1829:
1600:The reason this satisfies
1092:, applied to any dividend
26:
3675:
3517:
3161:{\displaystyle X^{2}+X+1}
2421:tenth-largest known prime
2034:{\displaystyle 1-\omega }
1974:of an Eisenstein integer
1672:. Any Eisenstein integer
3049:Surányi, László (1997).
3029:Dixon elliptic functions
1946:) which is congruent to
3775:1-systoles of manifolds
3751:Filling area conjecture
3057:Szalay, Mihály (1991).
2952:of real dimension
1597:-to-integer functions.
590:is given explicitly by
513:satisfies the equation
334:, in contrast with the
3734:1-systoles of surfaces
3686:Mathematics portal
3428:
3381:
3339:
3262:"Largest Known Primes"
3242:
3200:
3162:
3053:. TYPOTEX. p. 73.
2968:
2895:
2725:
2683:
2615:
2574:
2135:
2068:
2035:
1960:are equal to the norm
1866:
1858:
1804:
1728:
1587:
1558:
1409:
1196:
1079:
971:in the complex plane:
950:
883:
714:
561:
498:
364:algebraic number field
344:countably infinite set
323:
298:
198:
3429:
3382:
3340:
3243:
3201:
3163:
2966:
2938:of the complex plane
2896:
2726:
2684:
2616:
2575:
2136:
2069:
2036:
1864:
1840:
1805:
1729:
1588:
1559:
1410:
1197:
1080:
951:
884:
715:
562:
499:
321:
299:
199:
3761:Systoles of surfaces
3532:Constructible number
3391:
3349:
3307:
3210:
3172:
3133:
3078:"Eisenstein integer"
2746:
2693:
2629:
2584:
2470:
2082:
2045:
2019:
1738:
1694:
1571:
1439:
1257:
1239:unique factorization
1124:
1012:
967:formed by the sixth
963:in this ring is the
912:
734:
597:
520:
404:
230:
170:
76:"Eisenstein integer"
61:improve this article
3766:Eisenstein integers
3658:Supersilver ratio (
3649:Supergolden ratio (
1231:Euclidean algorithm
1105:, gives a quotient
149:Gotthold Eisenstein
145:Eisenstein integers
18:Eisenstein integers
3787:Essential manifold
3552:Eisenstein integer
3424:
3377:
3335:
3238:
3196:
3158:
3075:Weisstein, Eric W.
3019:Hurwitz quaternion
2969:
2891:
2778:
2721:
2679:
2611:
2570:
2502:
2131:
2064:
2031:
1867:
1859:
1800:
1769:
1724:
1709:
1583:
1554:
1501:
1471:
1405:
1394:
1374:
1311:
1192:
1090:division algorithm
1075:
946:
879:
824:
796:
727:, and is given by
710:
557:
494:
360:algebraic integers
328:triangular lattice
324:
313:cube root of unity
294:
194:
3864:Systolic geometry
3854:Cyclotomic fields
3844:Algebraic numbers
3831:
3830:
3823:Systolic category
3727:Systolic geometry
3693:
3692:
3667:Twelfth root of 2
3547:Doubling the cube
3537:Conway's constant
3522:Algebraic integer
3511:Algebraic numbers
3024:Quadratic integer
3009:Cubic reciprocity
2999:Systolic geometry
2973:Gaussian integers
2918:Eisenstein series
2889:
2833:
2794:
2749:
2666:
2557:
2518:
2473:
2454:Eisenstein series
2446:are congruent to
1826:Eisenstein primes
1768:
1764:
1708:
1704:
1621:quadratic integer
1530:
1527:
1523:
1519:
1516:
1500:
1499:
1470:
1469:
1393:
1392:
1373:
1372:
1355:
1349:
1334:
1328:
1324:
1310:
1288:
1278:
1272:
1268:
1161:
1158:
1154:
1150:
1147:
942:
924:
897:complex conjugate
875:
823:
795:
706:
553:
490:
394:is a root of the
336:Gaussian integers
311:(hence non-real)
265:
259:
153:Eulerian integers
137:
136:
129:
111:
16:(Redirected from
3876:
3818:Systolic freedom
3797:Hermite constant
3720:
3713:
3706:
3697:
3684:
3683:
3661:
3652:
3644:Square root of 7
3639:Square root of 6
3634:Square root of 5
3629:Square root of 3
3624:Square root of 2
3617:
3613:
3584:
3565:
3557:Gaussian integer
3542:Cyclotomic field
3504:
3497:
3490:
3481:
3462:
3461:
3459:
3458:
3444:
3438:
3437:
3433:
3431:
3430:
3425:
3423:
3422:
3418:
3386:
3384:
3383:
3378:
3361:
3360:
3344:
3342:
3341:
3336:
3319:
3318:
3299:
3293:
3292:
3285:
3279:
3278:
3276:
3275:
3258:
3252:
3251:
3247:
3245:
3244:
3239:
3237:
3205:
3203:
3202:
3197:
3186:
3185:
3180:
3168:is reducible in
3167:
3165:
3164:
3159:
3145:
3144:
3125:
3119:
3118:
3106:
3095:
3089:
3088:
3087:
3070:
3064:
3062:
3054:
3045:
3004:Hermite constant
2994:Cyclotomic field
2989:Gaussian integer
2978:
2955:
2943:
2929:
2915:
2906:
2900:
2898:
2897:
2892:
2890:
2888:
2887:
2886:
2873:
2872:
2871:
2859:
2844:
2839:
2835:
2834:
2829:
2818:
2808:
2807:
2795:
2793:
2792:
2780:
2777:
2764:
2737:
2730:
2728:
2727:
2722:
2720:
2688:
2686:
2685:
2680:
2672:
2668:
2667:
2662:
2651:
2641:
2640:
2620:
2618:
2617:
2612:
2610:
2609:
2605:
2579:
2577:
2576:
2571:
2563:
2559:
2558:
2553:
2542:
2532:
2531:
2519:
2517:
2516:
2504:
2501:
2488:
2465:
2461:
2449:
2445:
2441:
2434:, discovered by
2425:
2415:
2407:
2403:
2395:
2388:
2381:
2374:
2367:
2360:
2353:
2339:
2312:
2306:
2288:
2284:
2280:
2270:
2257:
2246:
2236:
2232:
2228:
2216:
2211:, ... (sequence
2151:
2140:
2138:
2137:
2132:
2127:
2126:
2073:
2071:
2070:
2065:
2063:
2062:
2040:
2038:
2037:
2032:
2014:
2006:
1983:
1973:
1959:
1955:
1949:
1926:
1920:
1914:
1908:
1898:
1892:
1886:
1880:
1874:
1856:
1848:
1832:Eisenstein ideal
1821:
1815:
1809:
1807:
1806:
1801:
1799:
1791:
1783:
1775:
1770:
1760:
1759:
1753:
1745:
1733:
1731:
1730:
1725:
1723:
1715:
1710:
1700:
1699:
1689:
1683:
1677:
1671:
1661:
1655:
1649:
1645:
1618:
1592:
1590:
1589:
1584:
1563:
1561:
1560:
1555:
1541:
1528:
1525:
1524:
1521:
1517:
1514:
1510:
1506:
1502:
1495:
1491:
1480:
1476:
1472:
1465:
1461:
1431:
1414:
1412:
1411:
1406:
1395:
1388:
1384:
1375:
1368:
1364:
1353:
1347:
1332:
1326:
1325:
1317:
1312:
1309:
1308:
1307:
1302:
1293:
1286:
1281:
1276:
1270:
1269:
1261:
1249:
1228:
1222:
1216:
1210:
1201:
1199:
1198:
1193:
1159:
1156:
1155:
1153: with
1152:
1148:
1145:
1116:
1111:and a remainder
1110:
1104:
1097:
1084:
1082:
1081:
1076:
1071:
1070:
1049:
1048:
1004:
997:Euclidean domain
991:Euclidean domain
986:
982:
955:
953:
952:
947:
940:
939:
938:
926:
925:
917:
904:
888:
886:
885:
880:
873:
872:
871:
850:
849:
835:
834:
825:
816:
810:
809:
804:
797:
788:
769:
768:
763:
762:
758:
719:
717:
716:
711:
704:
589:
579:
566:
564:
563:
558:
551:
532:
531:
512:
503:
501:
500:
495:
488:
487:
483:
482:
481:
460:
459:
416:
415:
396:monic polynomial
393:
378:cyclotomic field
375:
356:commutative ring
303:
301:
300:
295:
293:
292:
288:
266:
261:
260:
255:
240:
218:
212:
203:
201:
200:
195:
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
3884:
3883:
3879:
3878:
3877:
3875:
3874:
3873:
3834:
3833:
3832:
3827:
3806:Higher systoles
3801:
3770:
3746:Pu's inequality
3729:
3724:
3694:
3689:
3678:
3671:
3659:
3650:
3618:
3615:
3611:
3595:Rational number
3582:
3581:Plastic ratio (
3563:
3527:Chebyshev nodes
3513:
3508:
3471:
3466:
3465:
3456:
3454:
3446:
3445:
3441:
3400:
3389:
3388:
3352:
3347:
3346:
3310:
3305:
3304:
3301:
3300:
3296:
3287:
3286:
3282:
3273:
3271:
3260:
3259:
3255:
3208:
3207:
3175:
3170:
3169:
3136:
3131:
3130:
3127:
3126:
3122:
3115:
3104:
3097:
3096:
3092:
3073:
3072:
3071:
3067:
3056:
3048:
3046:
3042:
3037:
2985:
2976:
2953:
2939:
2932:
2925:
2914:
2908:
2902:
2878:
2874:
2863:
2845:
2819:
2813:
2809:
2799:
2784:
2744:
2743:
2735:
2691:
2690:
2689:if and only if
2652:
2646:
2642:
2632:
2627:
2626:
2587:
2582:
2581:
2543:
2537:
2533:
2523:
2508:
2468:
2467:
2463:
2459:
2456:
2447:
2443:
2439:
2432:Mersenne primes
2423:
2413:
2405:
2401:
2390:
2383:
2376:
2369:
2362:
2355:
2348:
2317:
2308:
2290:
2286:
2282:
2276:
2260:
2251:
2247:. For example:
2242:
2234:
2230:
2226:
2212:
2145:
2118:
2080:
2079:
2054:
2043:
2042:
2017:
2016:
2012:
1985:
1975:
1961:
1957:
1953:
1947:
1929:Gaussian primes
1922:
1916:
1910:
1900:
1894:
1888:
1882:
1876:
1870:
1850:
1842:
1835:
1828:
1817:
1811:
1810:. (The size of
1736:
1735:
1692:
1691:
1685:
1679:
1673:
1663:
1657:
1651:
1647:
1624:
1601:
1569:
1568:
1522: and
1488:
1484:
1452:
1448:
1437:
1436:
1419:
1297:
1285:
1255:
1254:
1245:
1233:, which proves
1224:
1218:
1212:
1206:
1122:
1121:
1112:
1106:
1099:
1093:
1062:
1040:
1010:
1009:
1000:
993:
984:
972:
930:
910:
909:
900:
863:
841:
826:
775:
743:
739:
737:
732:
731:
725:squared modulus
595:
594:
581:
571:
523:
518:
517:
508:
507:In particular,
473:
451:
450:
446:
407:
402:
401:
381:
366:
352:
338:, which form a
270:
241:
228:
227:
214:
208:
168:
167:
161:complex numbers
133:
122:
116:
113:
70:
68:
58:
46:
35:
32:
23:
22:
15:
12:
11:
5:
3882:
3880:
3872:
3871:
3866:
3861:
3859:Lattice points
3856:
3851:
3846:
3836:
3835:
3829:
3828:
3826:
3825:
3820:
3815:
3809:
3807:
3803:
3802:
3800:
3799:
3794:
3792:Filling radius
3789:
3784:
3778:
3776:
3772:
3771:
3769:
3768:
3763:
3758:
3753:
3748:
3743:
3737:
3735:
3731:
3730:
3725:
3723:
3722:
3715:
3708:
3700:
3691:
3690:
3676:
3673:
3672:
3670:
3669:
3664:
3655:
3646:
3641:
3636:
3631:
3626:
3621:
3614:
3610:Silver ratio (
3607:
3602:
3597:
3592:
3587:
3578:
3573:
3568:
3562:Golden ratio (
3559:
3554:
3549:
3544:
3539:
3534:
3529:
3524:
3518:
3515:
3514:
3509:
3507:
3506:
3499:
3492:
3484:
3478:
3477:
3470:
3469:External links
3467:
3464:
3463:
3439:
3421:
3417:
3413:
3410:
3407:
3403:
3399:
3396:
3376:
3373:
3370:
3367:
3364:
3359:
3355:
3334:
3331:
3328:
3325:
3322:
3317:
3313:
3294:
3280:
3253:
3236:
3233:
3229:
3226:
3221:
3218:
3215:
3195:
3192:
3189:
3184:
3179:
3157:
3154:
3151:
3148:
3143:
3139:
3120:
3113:
3107:. p. 77.
3090:
3065:
3039:
3038:
3036:
3033:
3032:
3031:
3026:
3021:
3016:
3011:
3006:
3001:
2996:
2991:
2984:
2981:
2931:
2922:
2912:
2885:
2881:
2877:
2870:
2866:
2862:
2858:
2854:
2851:
2848:
2842:
2838:
2832:
2828:
2825:
2822:
2816:
2812:
2806:
2802:
2798:
2791:
2787:
2783:
2776:
2773:
2770:
2767:
2763:
2759:
2756:
2752:
2740:gamma function
2719:
2716:
2712:
2709:
2704:
2701:
2698:
2678:
2675:
2671:
2665:
2661:
2658:
2655:
2649:
2645:
2639:
2635:
2608:
2604:
2600:
2597:
2594:
2590:
2569:
2566:
2562:
2556:
2552:
2549:
2546:
2540:
2536:
2530:
2526:
2522:
2515:
2511:
2507:
2500:
2497:
2494:
2491:
2487:
2483:
2480:
2476:
2455:
2452:
2412:not exceeding
2410:absolute value
2398:
2397:
2342:
2341:
2273:
2272:
2258:
2223:
2222:
2130:
2125:
2121:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2061:
2057:
2053:
2050:
2030:
2027:
2024:
2009:
2008:
1951:
1944:rational prime
1827:
1824:
1798:
1794:
1790:
1786:
1782:
1778:
1774:
1767:
1763:
1756:
1752:
1748:
1744:
1722:
1718:
1714:
1707:
1703:
1582:
1579:
1576:
1565:
1564:
1553:
1550:
1547:
1544:
1540:
1536:
1533:
1513:
1509:
1505:
1498:
1494:
1487:
1483:
1479:
1475:
1468:
1464:
1458:
1455:
1451:
1447:
1444:
1416:
1415:
1404:
1401:
1398:
1391:
1387:
1381:
1378:
1371:
1367:
1361:
1358:
1352:
1346:
1343:
1340:
1337:
1331:
1323:
1320:
1315:
1306:
1301:
1296:
1292:
1284:
1275:
1267:
1264:
1235:Euclid's lemma
1203:
1202:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1144:
1141:
1138:
1135:
1132:
1129:
1086:
1085:
1074:
1069:
1065:
1061:
1058:
1055:
1052:
1047:
1043:
1039:
1036:
1033:
1029:
1026:
1023:
1020:
1017:
992:
989:
969:roots of unity
961:group of units
957:
956:
945:
937:
933:
929:
923:
920:
890:
889:
878:
870:
866:
862:
859:
856:
853:
848:
844:
839:
833:
829:
822:
819:
813:
808:
803:
800:
794:
791:
785:
782:
779:
773:
767:
761:
757:
752:
749:
746:
742:
721:
720:
709:
703:
698:
695:
692:
689:
686:
683:
680:
677:
674:
671:
668:
665:
662:
659:
656:
653:
650:
647:
644:
641:
638:
633:
630:
627:
624:
619:
616:
611:
608:
605:
602:
568:
567:
556:
550:
547:
544:
541:
538:
535:
530:
526:
505:
504:
493:
486:
480:
476:
472:
469:
466:
463:
458:
454:
449:
445:
442:
437:
434:
431:
428:
425:
422:
419:
414:
410:
351:
348:
340:square lattice
305:
304:
291:
287:
283:
280:
277:
273:
269:
264:
258:
253:
250:
247:
244:
238:
235:
205:
204:
193:
190:
187:
184:
181:
178:
175:
157:Leonhard Euler
135:
134:
49:
47:
40:
33:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3881:
3870:
3867:
3865:
3862:
3860:
3857:
3855:
3852:
3850:
3847:
3845:
3842:
3841:
3839:
3824:
3821:
3819:
3816:
3814:
3811:
3810:
3808:
3804:
3798:
3795:
3793:
3790:
3788:
3785:
3783:
3780:
3779:
3777:
3773:
3767:
3764:
3762:
3759:
3757:
3756:Bolza surface
3754:
3752:
3749:
3747:
3744:
3742:
3739:
3738:
3736:
3732:
3728:
3721:
3716:
3714:
3709:
3707:
3702:
3701:
3698:
3688:
3687:
3682:
3674:
3668:
3665:
3663:
3656:
3654:
3647:
3645:
3642:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3608:
3606:
3603:
3601:
3600:Root of unity
3598:
3596:
3593:
3591:
3588:
3586:
3579:
3577:
3574:
3572:
3571:Perron number
3569:
3567:
3560:
3558:
3555:
3553:
3550:
3548:
3545:
3543:
3540:
3538:
3535:
3533:
3530:
3528:
3525:
3523:
3520:
3519:
3516:
3512:
3505:
3500:
3498:
3493:
3491:
3486:
3485:
3482:
3476:
3473:
3472:
3468:
3453:
3449:
3443:
3440:
3435:
3419:
3415:
3411:
3408:
3405:
3401:
3397:
3394:
3374:
3371:
3365:
3357:
3353:
3332:
3329:
3323:
3315:
3311:
3298:
3295:
3290:
3284:
3281:
3270:
3269:
3263:
3257:
3254:
3249:
3231:
3227:
3219:
3216:
3213:
3190:
3182:
3155:
3152:
3149:
3146:
3141:
3137:
3124:
3121:
3116:
3114:0-471-19079-9
3110:
3103:
3102:
3094:
3091:
3085:
3084:
3079:
3076:
3069:
3066:
3060:
3052:
3044:
3041:
3034:
3030:
3027:
3025:
3022:
3020:
3017:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2986:
2982:
2980:
2974:
2965:
2961:
2959:
2951:
2950:complex torus
2947:
2942:
2937:
2928:
2923:
2921:
2920:of weight 6.
2919:
2911:
2905:
2883:
2879:
2875:
2868:
2860:
2856:
2852:
2840:
2836:
2830:
2826:
2823:
2820:
2814:
2810:
2804:
2800:
2796:
2789:
2785:
2781:
2771:
2757:
2754:
2750:
2741:
2732:
2714:
2710:
2702:
2699:
2696:
2676:
2673:
2669:
2663:
2659:
2656:
2653:
2647:
2643:
2637:
2633:
2625:. In general
2624:
2621:is a root of
2606:
2602:
2598:
2595:
2592:
2588:
2567:
2564:
2560:
2554:
2550:
2547:
2544:
2538:
2534:
2528:
2524:
2520:
2513:
2509:
2505:
2495:
2481:
2478:
2474:
2453:
2451:
2437:
2433:
2429:
2424:10223 × 2 + 1
2422:
2417:
2411:
2394:
2387:
2380:
2373:
2366:
2359:
2352:
2347:
2346:
2345:
2337:
2333:
2329:
2325:
2321:
2316:
2315:
2314:
2311:
2305:
2301:
2297:
2293:
2279:
2268:
2264:
2259:
2255:
2250:
2249:
2248:
2245:
2240:
2220:
2215:
2210:
2206:
2202:
2198:
2194:
2190:
2186:
2182:
2178:
2174:
2170:
2166:
2162:
2158:
2155:
2154:
2153:
2149:
2142:
2123:
2119:
2115:
2112:
2103:
2100:
2094:
2091:
2088:
2085:
2077:
2059:
2055:
2051:
2048:
2028:
2025:
2022:
2004:
2001:
1997:
1993:
1989:
1982:
1978:
1972:
1968:
1964:
1952:
1945:
1941:
1937:
1936:
1935:
1932:
1930:
1925:
1919:
1913:
1907:
1903:
1897:
1891:
1885:
1879:
1873:
1863:
1854:
1846:
1839:
1833:
1825:
1823:
1820:
1814:
1792:
1784:
1776:
1765:
1761:
1754:
1746:
1716:
1705:
1701:
1688:
1682:
1676:
1670:
1666:
1660:
1654:
1644:
1641:
1637:
1634:
1630:
1627:
1622:
1616:
1612:
1608:
1604:
1598:
1596:
1577:
1551:
1548:
1545:
1542:
1538:
1534:
1531:
1511:
1507:
1503:
1496:
1492:
1485:
1481:
1477:
1473:
1466:
1462:
1456:
1453:
1449:
1445:
1442:
1435:
1434:
1433:
1430:
1426:
1422:
1418:for rational
1402:
1399:
1396:
1389:
1385:
1379:
1376:
1369:
1365:
1359:
1356:
1350:
1344:
1341:
1338:
1335:
1329:
1318:
1313:
1304:
1294:
1282:
1273:
1265:
1262:
1253:
1252:
1251:
1248:
1242:
1240:
1236:
1232:
1227:
1221:
1215:
1209:
1189:
1183:
1177:
1174:
1168:
1162:
1142:
1139:
1136:
1133:
1130:
1127:
1120:
1119:
1118:
1115:
1109:
1102:
1096:
1091:
1072:
1067:
1063:
1059:
1056:
1053:
1050:
1045:
1041:
1037:
1031:
1027:
1024:
1021:
1015:
1008:
1007:
1006:
1003:
998:
990:
988:
980:
976:
970:
966:
962:
943:
935:
931:
927:
918:
908:
907:
906:
903:
898:
893:
876:
868:
864:
860:
857:
854:
851:
846:
842:
837:
831:
827:
820:
817:
811:
806:
798:
792:
789:
783:
780:
771:
765:
759:
755:
750:
747:
744:
740:
730:
729:
728:
726:
707:
701:
693:
690:
687:
684:
681:
678:
675:
672:
666:
660:
657:
654:
651:
648:
642:
636:
631:
628:
625:
614:
609:
606:
603:
593:
592:
591:
588:
584:
578:
574:
554:
548:
545:
542:
539:
536:
533:
528:
524:
516:
515:
514:
511:
491:
484:
478:
474:
470:
467:
464:
461:
456:
452:
447:
443:
440:
432:
429:
426:
423:
417:
412:
408:
400:
399:
398:
397:
392:
388:
384:
379:
373:
369:
365:
361:
357:
349:
347:
345:
341:
337:
333:
332:complex plane
329:
320:
316:
314:
310:
289:
285:
281:
278:
275:
271:
267:
262:
256:
251:
248:
245:
242:
236:
233:
226:
225:
224:
222:
217:
211:
191:
188:
185:
182:
179:
176:
173:
166:
165:
164:
162:
158:
154:
150:
147:(named after
146:
142:
131:
128:
120:
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
30:
19:
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1980:
1976:
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1966:
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1943:
1940:prime number
1938:an ordinary
1933:
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1911:
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1895:
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1883:
1877:
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1213:
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978:
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965:cyclic group
958:
901:
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586:
582:
576:
572:
569:
509:
506:
390:
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382:
376:– the third
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163:of the form
152:
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59:Please help
54:verification
51:
3452:fungrim.org
3303:"Show that
3268:Prime Pages
3059:Számelmélet
2623:j-invariant
2252:3 = −(1 + 2
999:whose norm
159:), are the
141:mathematics
3838:Categories
3457:2023-06-22
3274:2023-02-27
2076:associates
1899:such that
905:satisfies
895:Also, the
350:Properties
87:newspapers
3409:π
3395:ρ
3372:≠
3366:ρ
3330:≠
3217:≡
3083:MathWorld
2880:π
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2824:π
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2758:∈
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2428:PrimeGrid
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2120:ω
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117:July 2020
3869:Integers
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221:integers
3051:Algebra
2946:lattice
2944:by the
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2448:1 mod 3
2440:2 mod 3
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362:in the
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