3382:
190:
2741:
2733:
1247:
1551:
A quadratic integer has the same norm as its conjugate, and this norm is the product of the quadratic integer and its conjugate. The conjugate of a sum or a product of quadratic integers is the sum or the product (respectively) of the conjugates. This means that the conjugation is an
1107:
3044:
2907:
895:
3107:
3826:
3618:
3192:
2146:
2088:
1810:
1546:
1062:
3674:, for which the ring of quadratic integers is a principal ideal ring. However, the complete list is not known; it is not even known if the number of these principal ideal rings is finite or not.
1737:
3775:
3367:
3321:
731:
331:
1899:
1663:
789:
581:
2397:
2275:
2213:
1850:
1590:
1348:
937:
535:
430:
685:
2508:
1296:
990:
2452:
3261:
3228:
621:
4238:
2805:
452:
4122:
645:
3907:
quadratic integers that is a principal ideal domain is also a
Euclidean domain for some Euclidean function, which can indeed differ from the usual norm. The values
1242:{\displaystyle \omega ={\begin{cases}{\sqrt {D}}&{\mbox{if }}D\equiv 2,3{\pmod {4}}\\{{1+{\sqrt {D}}} \over 2}&{\mbox{if }}D\equiv 1{\pmod {4}}\end{cases}}}
2475:
3373:, as a simple computation shows that their product is the ideal generated by 3, and, if they were principal, this would imply that 3 would not be irreducible.
3267:, as they have all a norm of 9, and if they were not irreducible, they would have a factor of norm 3, which is impossible, the norm of an element different of
2514:
fundamental unit" the unique one which has an absolute value greater than 1 (as a real number). It is the unique fundamental unit that may be written as
3867:
2967:
2821:
829:
3896:
the four corresponding rings of quadratic integers are among the rare known examples of principal ideal domains that are not
Euclidean domains.
4277:
4103:
1454:
3059:
2014:
734:
3784:
3576:
3115:
2104:
2046:
1768:
1487:
4323:
2416:
1011:
4145:
2025:. Being the simplest examples of algebraic integers, quadratic integers are commonly the starting examples of most studies of
3900:
3477:
1695:
3698:
3326:
3280:
2156:
4180:
3555:
3050:
690:
3573:
The imaginary rings of quadratic integers that are principal ideal rings have been completely determined. These are
267:
3874:
There is no other ring of quadratic integers that is
Euclidean with the norm as a Euclidean function. For negative
3656:
1865:
1629:
755:
547:
2366:
2244:
2182:
1819:
1559:
1317:
906:
504:
399:
2760:
or otherwise non-real) number. Therefore, it is natural to treat a quadratic integer ring as a set of algebraic
650:
3914:
were the first for which the ring of quadratic integers was proven to be
Euclidean, but not norm-Euclidean.
2026:
174:
3559:
2224:
738:
3688:
When a ring of quadratic integers is a principal ideal domain, it is interesting to know whether it is a
593:
quadratic integers is not a ring because it is not closed under addition or multiplication. For example,
3404:, and the corresponding quadratic integer ring is a set of algebraic real numbers. The solutions of the
2480:
953:
2427:
1259:
4328:
3661:
3541:
3419:
2954:
1908:
1457:, which means that the norm of a product of quadratic integers is always the product of their norms.
1073:
158:
3233:
3200:
1122:
3492:
3275:
3264:
2018:
1607:
596:
455:
4208:
2771:
435:
4263:
4095:
3879:
3778:
3643:
3423:
2910:
2306:
2176:
626:
586:
221:
151:
3544:
property is not always verified for rings of quadratic integers, as seen above for the case of
3405:
229:
162:
4333:
4273:
4157:
4099:
3563:
3381:
2017:
is not true in many rings of quadratic integers. However, there is a unique factorization for
1623:
368:
258:
79:(usual) integers. When algebraic integers are considered, the usual integers are often called
47:
4283:
4252:
4176:
4137:
4087:
3689:
3683:
2958:
2918:
2808:
2757:
2278:
2228:
1853:
1813:
242:
116:
4113:
4041:. In: Analytic Number Theory (St. Louis, 1972), Proc. Sympos. Pure Math. 24(1973), 321β332.
4287:
4109:
3567:
3551:
3370:
2947:
2022:
1619:
750:
538:
501:
The quadratic integers (including the ordinary integers) that belong to a quadratic field
394:
364:
43:
2457:
4267:
3496:
2761:
1979:
The square root of any integer is a quadratic integer, as every integer can be written
1443:
262:
166:
100:
3271:
being at least 4. Thus the factorization of 9 into irreducible factors is not unique.
189:
4317:
3528:
1303:
124:
31:
4305:
3651:
3647:
3473:
2740:
2592:
1927:
1553:
1391:
749:
Here and in the following, the quadratic integers that are considered belong to a
17:
795:
is a square-free integer. This does not restrict the generality, as the equality
3503:
3401:
3039:{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}}\,)}=\mathbf {Z} \left,}
2684:
120:
87:
2902:{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-3}}\,)}=\mathbf {Z} \left}
3635:
2812:
890:{\displaystyle \mathbb {Q} ({\sqrt {D}}\,)=\mathbb {Q} ({\sqrt {a^{2}D}}\,).}
2021:, which is expressed by the fact that every ring of algebraic integers is a
585:
Although the quadratic integers belonging to a given quadratic field form a
4256:
3955:
2032:
The quadratic integer rings divide in two classes depending on the sign of
224:
had already discovered a multiplication of quadratic integers of the same
3878:, a ring of quadratic integers is Euclidean if and only if the norm is a
4158:"Algebraic theory of Penrose's non-periodic tilings of the plane, I, II"
2732:
4141:
360:
39:
3491:
is an arbitrary integer. This ring also arises from studying 5-fold
2159:β which measures the failure of unique factorization β is given in
3639:
3380:
2739:
2731:
2543:
The fundamental units for the 10 smallest positive square-free
3102:{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}}\,)},}
184:
27:
Root of a quadratic polynomial with a unit leading coefficient
4123:"A quadratic field which is Euclidean but not norm-Euclidean"
939:
is a quadratic integer if and only if there are two integers
3821:{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {D}}\,)}}
3791:
3613:{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {D}}\,)}}
3583:
3187:{\displaystyle 9=3\cdot 3=(2+{\sqrt {-5}})(2-{\sqrt {-5}}).}
3066:
2974:
2828:
2687:
of the fundamental unit may be very large. For example, for
2141:{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {D}}\,)}}
2111:
2083:{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {D}}\,)}}
2053:
1805:{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {D}}\,)}}
1775:
1541:{\displaystyle {\overline {a+b{\sqrt {D}}}}=a-b{\sqrt {D}}.}
50:
of degree two, that is, solutions of equations of the form
3862:
3858:= 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73
2164:
2160:
1996:
is a square-free integer, and its square root is a root of
1235:
1057:{\displaystyle x={\frac {a}{2}}+{\frac {b}{2}}{\sqrt {D}},}
4182:
2424:, there are three other fundamental units, its conjugate
1435:
The norm of a quadratic integer is always an integer. If
4309:, Version 3.01, September 28, 2008. online lecture notes
3527:. Some results were presented to European community by
1442:, the norm of a quadratic integer is the square of its
1079:
In other words, every quadratic integer may be written
363:. Each quadratic integer that is not an integer is not
201:
3692:. This problem has been completely solved as follows.
1706:
1262:
1201:
1135:
4211:
3787:
3701:
3579:
3329:
3283:
3236:
3203:
3118:
3062:
2970:
2824:
2815:
around 1800 to state his biquadratic reciprocity law.
2774:
2483:
2460:
2430:
2369:
2247:
2185:
2107:
2049:
1868:
1822:
1771:
1698:
1632:
1562:
1490:
1320:
1110:
1014:
956:
909:
832:
758:
693:
653:
629:
599:
550:
507:
438:
402:
270:
3956:"Why is quadratic integer ring defined in that way?"
2917:
Both rings mentioned above are rings of integers of
1732:{\displaystyle \omega ={\tfrac {1+{\sqrt {D}}}{2}}}
4232:
3820:
3770:{\displaystyle N(a+b{\sqrt {D}}\,)=|a^{2}-Db^{2}|}
3769:
3650:, although Heegner's proof was not believed until
3612:
3362:{\displaystyle \langle 3,1-{\sqrt {-5}}\,\rangle }
3361:
3316:{\displaystyle \langle 3,1+{\sqrt {-5}}\,\rangle }
3315:
3255:
3222:
3186:
3101:
3038:
2901:
2799:
2502:
2469:
2446:
2391:
2269:
2207:
2140:
2082:
1893:
1844:
1804:
1731:
1657:
1584:
1540:
1342:
1290:
1241:
1056:
984:
931:
889:
783:
725:
679:
639:
615:
575:
529:
446:
424:
325:
157:Quadratic integers occur in the solutions of many
726:{\displaystyle (1+{\sqrt {2}})\cdot {\sqrt {3}}}
487:is positive, the quadratic integer is real. If
326:{\displaystyle x=(-b\pm {\sqrt {b^{2}-4c}})/2}
86:Common examples of quadratic integers are the
3684:Euclidean domain Β§ Norm-Euclidean fields
2148:are the ordinary integers, and the ring is a
241:of the quadratic integers was first given by
8:
3356:
3330:
3310:
3284:
2399:has infinitely many units that are equal to
2163:; for the imaginary case, they are given in
1894:{\displaystyle \mathbf {Q} ({\sqrt {D}}\,).}
1658:{\displaystyle \mathbf {Q} ({\sqrt {D}}\,).}
1594:
784:{\displaystyle \mathbb {Q} ({\sqrt {D}}\,),}
576:{\displaystyle \mathbb {Q} ({\sqrt {D}}\,).}
237:
228:, which allowed them to solve some cases of
4013:
3990:
2728:Examples of complex quadratic integer rings
2540:positive (integers or halves of integers).
2392:{\displaystyle \mathbb {Q} ({\sqrt {D}}\,)}
2270:{\displaystyle \mathbb {Q} ({\sqrt {D}}\,)}
2208:{\displaystyle \mathbb {Q} ({\sqrt {D}}\,)}
1943:by the square of an integer. In particular
1845:{\displaystyle \mathbf {Q} ({\sqrt {D}}\,)}
1585:{\displaystyle \mathbb {Q} ({\sqrt {D}}\,)}
1343:{\displaystyle \mathbb {Q} ({\sqrt {D}}\,)}
932:{\displaystyle \mathbb {Q} ({\sqrt {D}}\,)}
530:{\displaystyle \mathbb {Q} ({\sqrt {D}}\,)}
454:generated by the square root of the unique
425:{\displaystyle \mathbb {Q} ({\sqrt {D}}\,)}
3628:= β1, β2, β3, β7, β11, β19, β43, β67, β163
2694:, the fundamental units are respectively
2277:has at most six units. In the case of the
165:, and other questions related to integral
4269:Topics in Number Theory, Volumes I and II
4220:
4213:
4212:
4210:
4002:
3812:
3805:
3797:
3796:
3790:
3789:
3786:
3762:
3756:
3740:
3731:
3724:
3717:
3700:
3604:
3597:
3589:
3588:
3582:
3581:
3578:
3355:
3345:
3328:
3309:
3299:
3282:
3243:
3235:
3210:
3202:
3168:
3146:
3117:
3090:
3080:
3072:
3071:
3065:
3064:
3061:
3027:
3017:
3007:
2998:
2988:
2980:
2979:
2973:
2972:
2969:
2879:
2872:
2870:
2861:
2852:
2842:
2834:
2833:
2827:
2826:
2823:
2793:
2783:
2775:
2773:
2487:
2482:
2459:
2431:
2429:
2385:
2378:
2371:
2370:
2368:
2263:
2256:
2249:
2248:
2246:
2201:
2194:
2187:
2186:
2184:
2132:
2125:
2117:
2116:
2110:
2109:
2106:
2074:
2067:
2059:
2058:
2052:
2051:
2048:
1884:
1877:
1869:
1867:
1838:
1831:
1823:
1821:
1796:
1789:
1781:
1780:
1774:
1773:
1770:
1715:
1705:
1697:
1648:
1641:
1633:
1631:
1578:
1571:
1564:
1563:
1561:
1528:
1503:
1491:
1489:
1336:
1329:
1322:
1321:
1319:
1298:is impossible, since it would imply that
1272:
1261:
1216:
1200:
1185:
1178:
1176:
1156:
1134:
1125:
1117:
1109:
1044:
1034:
1021:
1013:
972:
955:
925:
918:
911:
910:
908:
880:
869:
863:
856:
855:
848:
841:
834:
833:
831:
774:
767:
760:
759:
757:
716:
703:
692:
680:{\displaystyle 1+{\sqrt {2}}+{\sqrt {3}}}
670:
660:
652:
630:
628:
606:
598:
566:
559:
552:
551:
549:
523:
516:
509:
508:
506:
440:
439:
437:
418:
411:
404:
403:
401:
315:
295:
289:
269:
4039:On Euclidean rings of algebraic integers
3942:
3930:
3660:). This is a special case of the famous
3377:Examples of real quadratic integer rings
2961:for the norm. This is not the case for
261:of degree two. More explicitly, it is a
4025:
3923:
3667:There are many known positive integers
1256:has been supposed square-free the case
333:, which solves an equation of the form
4092:Elements of the history of mathematics
4050:
3422:that has been widely studied, are the
2345:, there are only two units, which are
2155:For real quadratic integer rings, the
1446:as a complex number (this is false if
1383:are either both integers, or, only if
4061:
3678:Euclidean rings of quadratic integers
3536:Principal rings of quadratic integers
2231:of its conjugate in the second case.
7:
4196:Dummit, D. S.; Foote, R. M. (2004),
3554:, a ring of quadratic integers is a
393:β and lies in a uniquely determined
119:. Another common example is the non-
3828:is a Euclidean domain for negative
1280:
1273:
1224:
1217:
1164:
1157:
1455:completely multiplicative function
38:are a generalization of the usual
25:
3978:
3562:. This occurs if and only if the
3521:, corresponding to the ring is
3495:on Euclidean plane, for example,
2503:{\displaystyle -{\overline {u}}.}
2015:fundamental theorem of arithmetic
1291:{\textstyle D\equiv 0{\pmod {4}}}
985:{\displaystyle x=a+b{\sqrt {D}},}
498:(that is, complex and non-real).
4272:. New York: Dover Publications.
3798:
3654:gave a later proof in 1967 (see
3590:
3073:
3053:. This can be shown as follows.
3008:
2981:
2942:) correspondingly. In contrast,
2862:
2835:
2776:
2447:{\displaystyle {\overline {u}},}
2118:
2060:
1958:, being a root of the equation
1870:
1824:
1782:
1634:
1398:of such a quadratic integer is
188:
2179:in the ring of the integers of
1765:otherwise. It is often denoted
1556:of the ring of the integers of
173:is basic for many questions of
4227:
4217:
3901:generalized Riemann hypothesis
3882:for it. It follows that, for
3813:
3802:
3763:
3732:
3725:
3705:
3605:
3594:
3478:Peter Gustav Lejeune Dirichlet
3256:{\displaystyle 2-{\sqrt {-5}}}
3223:{\displaystyle 2+{\sqrt {-5}}}
3178:
3159:
3156:
3137:
3091:
3077:
2999:
2985:
2853:
2839:
2794:
2780:
2414:is a particular unit called a
2386:
2375:
2363:, the ring of the integers of
2264:
2253:
2241:, the ring of the integers of
2202:
2191:
2150:complex quadratic integer ring
2133:
2122:
2075:
2064:
1885:
1874:
1839:
1828:
1797:
1786:
1649:
1638:
1595:Β§ Quadratic integer rings
1579:
1568:
1337:
1326:
1284:
1274:
1228:
1218:
1168:
1158:
926:
915:
881:
860:
849:
838:
775:
764:
710:
694:
567:
556:
524:
513:
419:
408:
312:
277:
238:Β§ Explicit representation
235:The characterization given in
90:of rational integers, such as
1:
2410:is an arbitrary integer, and
616:{\displaystyle 1+{\sqrt {2}}}
4233:{\displaystyle \mathbb {Z} }
3506:treated the Pell's equation
2800:{\displaystyle \mathbf {Z} }
2492:
2436:
2227:is its conjugate. It is the
2101:, the only real elements of
2090:are real, and the ring is a
1511:
647:are quadratic integers, but
447:{\displaystyle \mathbb {Q} }
4098:. Berlin: Springer-Verlag.
3556:unique factorization domain
3476:. This ring was studied by
3051:unique factorization domain
2420:. Given a fundamental unit
2215:if and only if its norm is
2092:real quadratic integer ring
640:{\displaystyle {\sqrt {3}}}
171:rings of quadratic integers
4350:
3681:
3480:. Its units have the form
3385:Powers of the golden ratio
2811:, which was introduced by
822:(for any positive integer
46:. Quadratic integers are
4165:Indagationes Mathematicae
4156:de Bruijn, N. G. (1981),
3400:is a positive irrational
2341:. For all other negative
2175:A quadratic integer is a
1610:(different from 0 and 1)
4121:Clark, David A. (1994),
3550:. However, as for every
2953:Both above examples are
2223:. In the first case its
1460:Every quadratic integer
1097:are integers, and where
4324:Algebraic number theory
4306:Algebraic Number Theory
4130:Manuscripta Mathematica
4014:Dummit & Foote 2004
3991:Dummit & Foote 2004
3903:implies that a ring of
3899:On the other hand, the
3695:Equipped with the norm
3558:if and only if it is a
2027:algebraic number theory
1602:Quadratic integer rings
1314:A quadratic integer in
745:Explicit representation
175:algebraic number theory
4257:10.4153/CJM-2004-003-9
4234:
4185:(2nd ed.), Vieweg
3960:math.stackexchange.com
3933:, Supplement X, p. 447
3822:
3771:
3634:This result was first
3614:
3560:principal ideal domain
3386:
3363:
3317:
3257:
3224:
3188:
3103:
3040:
2903:
2801:
2745:
2737:
2504:
2471:
2448:
2393:
2288:), the four units are
2271:
2225:multiplicative inverse
2209:
2142:
2084:
1976:as its discriminant.
1895:
1846:
1806:
1733:
1659:
1616:quadratic integer ring
1586:
1542:
1392:halves of odd integers
1344:
1292:
1243:
1058:
986:
933:
891:
785:
727:
681:
641:
617:
577:
531:
448:
426:
367:β namely, it's a real
327:
150:, which generates the
115:, which generates the
4235:
3890:= β19, β43, β67, β163
3840:= β1,ββ2,ββ3,ββ7,ββ11
3823:
3772:
3657:StarkβHeegner theorem
3615:
3566:of the corresponding
3502:Indian mathematician
3384:
3364:
3318:
3258:
3225:
3189:
3104:
3041:
2955:principal ideal rings
2904:
2802:
2768:A classic example is
2743:
2735:
2510:Commonly, one calls "
2505:
2472:
2449:
2394:
2316:), the six units are
2305:. In the case of the
2272:
2210:
2143:
2085:
1896:
1847:
1812:, because it is the
1807:
1734:
1660:
1587:
1543:
1345:
1293:
1244:
1059:
987:
934:
892:
786:
728:
682:
642:
618:
578:
532:
449:
427:
328:
222:Indian mathematicians
159:Diophantine equations
4209:
4205:Harper, M. (2004), "
4028:, pp. II:57, 81
3785:
3699:
3662:class number problem
3577:
3542:unique factorization
3426:of these rings, for
3420:Diophantine equation
3327:
3281:
3234:
3201:
3116:
3060:
3049:which is not even a
2968:
2822:
2772:
2481:
2458:
2428:
2367:
2245:
2183:
2105:
2047:
1866:
1820:
1769:
1696:
1630:
1560:
1488:
1318:
1310:Norm and conjugation
1302:is divisible by the
1260:
1108:
1012:
954:
907:
830:
756:
691:
651:
627:
597:
548:
505:
436:
400:
268:
4264:LeVeque, William J.
3493:rotational symmetry
2911:Eisenstein integers
2752: < 0,
2307:Eisenstein integers
1608:square-free integer
735:minimal polynomials
543:ring of integers of
456:square-free integer
432:, the extension of
152:Eisenstein integers
4230:
4142:10.1007/BF02567617
3880:Euclidean function
3846:and, for positive
3818:
3779:Euclidean function
3767:
3610:
3387:
3359:
3313:
3253:
3220:
3184:
3099:
3036:
2899:
2797:
2746:
2738:
2500:
2470:{\displaystyle -u}
2467:
2444:
2389:
2267:
2205:
2138:
2080:
2043:, all elements of
1939:is the product of
1891:
1842:
1802:
1729:
1727:
1655:
1624:algebraic integers
1622:consisting of the
1582:
1538:
1340:
1288:
1239:
1234:
1205:
1139:
1054:
982:
947:such that either
929:
887:
781:
733:are not, as their
723:
677:
637:
613:
573:
527:
444:
422:
323:
200:. You can help by
48:algebraic integers
36:quadratic integers
18:Quadratic integers
4279:978-0-486-42539-9
4225:
4177:Dedekind, Richard
4105:978-3-540-64767-6
4088:Bourbaki, Nicolas
3810:
3722:
3602:
3353:
3307:
3251:
3218:
3176:
3154:
3088:
3025:
2996:
2959:Euclidean domains
2919:cyclotomic fields
2893:
2887:
2850:
2809:Gaussian integers
2791:
2744:Eisenstein primes
2736:Gaussian integers
2495:
2439:
2383:
2279:Gaussian integers
2261:
2199:
2130:
2072:
1911:of all equations
1882:
1836:
1794:
1726:
1720:
1646:
1576:
1533:
1514:
1508:
1453:). The norm is a
1334:
1204:
1196:
1190:
1138:
1130:
1049:
1042:
1029:
1002:is a multiple of
977:
923:
878:
846:
772:
721:
708:
675:
665:
635:
611:
564:
521:
475:for some integer
416:
369:irrational number
310:
259:algebraic integer
255:quadratic integer
218:
217:
117:Gaussian integers
81:rational integers
16:(Redirected from
4341:
4291:
4259:
4245:Can. J. Math. 56
4239:
4237:
4236:
4231:
4226:
4221:
4216:
4201:
4198:Abstract Algebra
4192:
4191:
4190:
4172:
4162:
4152:
4150:
4144:, archived from
4127:
4117:
4094:. Translated by
4083:
4064:
4059:
4053:
4048:
4042:
4035:
4029:
4023:
4017:
4011:
4005:
4000:
3994:
3988:
3982:
3976:
3970:
3969:
3967:
3966:
3952:
3946:
3940:
3934:
3928:
3913:
3891:
3877:
3865:
3859:
3849:
3841:
3831:
3827:
3825:
3824:
3819:
3817:
3816:
3811:
3806:
3801:
3795:
3794:
3776:
3774:
3773:
3768:
3766:
3761:
3760:
3745:
3744:
3735:
3723:
3718:
3690:Euclidean domain
3673:
3629:
3619:
3617:
3616:
3611:
3609:
3608:
3603:
3598:
3593:
3587:
3586:
3549:
3526:
3520:
3518:
3512:
3490:
3486:
3471:
3470:
3468:
3467:
3464:
3461:
3460:
3459:
3443:
3432:
3417:
3399:
3395:
3368:
3366:
3365:
3360:
3354:
3346:
3322:
3320:
3319:
3314:
3308:
3300:
3270:
3262:
3260:
3259:
3254:
3252:
3244:
3229:
3227:
3226:
3221:
3219:
3211:
3193:
3191:
3190:
3185:
3177:
3169:
3155:
3147:
3108:
3106:
3105:
3100:
3095:
3094:
3089:
3081:
3076:
3070:
3069:
3045:
3043:
3042:
3037:
3032:
3028:
3026:
3018:
3011:
3003:
3002:
2997:
2989:
2984:
2978:
2977:
2908:
2906:
2905:
2900:
2898:
2894:
2889:
2888:
2880:
2871:
2865:
2857:
2856:
2851:
2843:
2838:
2832:
2831:
2818:The elements in
2806:
2804:
2803:
2798:
2792:
2784:
2779:
2755:
2751:
2723:
2722:
2721:
2713:
2712:
2711:
2703:
2702:
2701:
2693:
2682:
2678:
2677:
2676:
2668:
2667:
2666:
2658:
2657:
2655:
2654:
2651:
2648:
2647:
2646:
2634:
2633:
2632:
2624:
2623:
2622:
2614:
2613:
2612:
2604:
2603:
2602:
2590:
2589:
2587:
2586:
2583:
2580:
2579:
2578:
2566:
2565:
2564:
2556:
2555:
2554:
2546:
2539:
2535:
2531:
2530:
2529:
2509:
2507:
2506:
2501:
2496:
2488:
2476:
2474:
2473:
2468:
2453:
2451:
2450:
2445:
2440:
2432:
2423:
2417:fundamental unit
2413:
2409:
2405:
2398:
2396:
2395:
2390:
2384:
2379:
2374:
2362:
2352:
2348:
2344:
2340:
2339:
2337:
2336:
2333:
2330:
2329:
2328:
2315:
2304:
2303:
2302:
2296:
2295:
2287:
2276:
2274:
2273:
2268:
2262:
2257:
2252:
2240:
2222:
2218:
2214:
2212:
2211:
2206:
2200:
2195:
2190:
2147:
2145:
2144:
2139:
2137:
2136:
2131:
2126:
2121:
2115:
2114:
2100:
2089:
2087:
2086:
2081:
2079:
2078:
2073:
2068:
2063:
2057:
2056:
2042:
2035:
2009:
1995:
1991:
1975:
1968:
1957:
1951:
1950:
1942:
1938:
1925:
1907:consists of all
1906:
1900:
1898:
1897:
1892:
1883:
1878:
1873:
1861:
1854:integral closure
1851:
1849:
1848:
1843:
1837:
1832:
1827:
1814:ring of integers
1811:
1809:
1808:
1803:
1801:
1800:
1795:
1790:
1785:
1779:
1778:
1764:
1763:
1762:
1749:
1738:
1736:
1735:
1730:
1728:
1722:
1721:
1716:
1707:
1691:
1664:
1662:
1661:
1656:
1647:
1642:
1637:
1613:
1591:
1589:
1588:
1583:
1577:
1572:
1567:
1547:
1545:
1544:
1539:
1534:
1529:
1515:
1510:
1509:
1504:
1492:
1477:
1476:
1475:
1452:
1441:
1430:
1421:
1420:
1389:
1382:
1378:
1370:
1369:
1368:
1350:may be written
1349:
1347:
1346:
1341:
1335:
1330:
1325:
1301:
1297:
1295:
1294:
1289:
1287:
1255:
1248:
1246:
1245:
1240:
1238:
1237:
1231:
1206:
1202:
1197:
1192:
1191:
1186:
1177:
1171:
1140:
1136:
1131:
1126:
1100:
1096:
1092:
1088:
1071:
1067:
1063:
1061:
1060:
1055:
1050:
1045:
1043:
1035:
1030:
1022:
1005:
1001:
991:
989:
988:
983:
978:
973:
946:
942:
938:
936:
935:
930:
924:
919:
914:
902:
896:
894:
893:
888:
879:
874:
873:
864:
859:
847:
842:
837:
825:
821:
820:
819:
807:
806:
794:
790:
788:
787:
782:
773:
768:
763:
732:
730:
729:
724:
722:
717:
709:
704:
686:
684:
683:
678:
676:
671:
666:
661:
646:
644:
643:
638:
636:
631:
622:
620:
619:
614:
612:
607:
582:
580:
579:
574:
565:
560:
555:
536:
534:
533:
528:
522:
517:
512:
493:
486:
480:
474:
460:
453:
451:
450:
445:
443:
431:
429:
428:
423:
417:
412:
407:
392:
382:and non-real if
381:
359:
353:
347:
332:
330:
329:
324:
319:
311:
300:
299:
290:
243:Richard Dedekind
227:
213:
210:
192:
185:
163:Pell's equations
149:
148:
146:
145:
142:
139:
138:
137:
114:
113:
112:
98:
97:
96:
78:
74:
67:
44:quadratic fields
21:
4349:
4348:
4344:
4343:
4342:
4340:
4339:
4338:
4314:
4313:
4312:
4299:
4297:Further reading
4294:
4280:
4262:
4207:
4206:
4204:
4195:
4188:
4186:
4175:
4160:
4155:
4148:
4125:
4120:
4106:
4086:
4077:
4073:
4068:
4067:
4060:
4056:
4049:
4045:
4037:P. Weinberger,
4036:
4032:
4024:
4020:
4012:
4008:
4001:
3997:
3989:
3985:
3977:
3973:
3964:
3962:
3954:
3953:
3949:
3941:
3937:
3929:
3925:
3920:
3908:
3886:
3875:
3861:
3854:
3847:
3836:
3829:
3788:
3783:
3782:
3752:
3736:
3697:
3696:
3686:
3680:
3668:
3624:
3580:
3575:
3574:
3568:quadratic field
3552:Dedekind domain
3545:
3538:
3522:
3514:
3508:
3507:
3497:Penrose tilings
3488:
3481:
3465:
3462:
3457:
3455:
3453:
3452:
3450:
3445:
3438:
3427:
3408:
3406:Pell's equation
3397:
3390:
3379:
3325:
3324:
3279:
3278:
3268:
3232:
3231:
3199:
3198:
3197:The factors 3,
3114:
3113:
3063:
3058:
3057:
3016:
3012:
2971:
2966:
2965:
2948:Dedekind domain
2941:
2930:
2866:
2825:
2820:
2819:
2770:
2769:
2762:complex numbers
2753:
2749:
2730:
2719:
2717:
2715:
2709:
2707:
2705:
2699:
2697:
2695:
2688:
2680:
2674:
2672:
2670:
2664:
2662:
2660:
2652:
2649:
2644:
2642:
2640:
2639:
2637:
2636:
2630:
2628:
2626:
2620:
2618:
2616:
2610:
2608:
2606:
2600:
2598:
2596:
2584:
2581:
2576:
2574:
2572:
2571:
2569:
2568:
2562:
2560:
2558:
2552:
2550:
2548:
2544:
2537:
2533:
2525:
2523:
2515:
2479:
2478:
2456:
2455:
2426:
2425:
2421:
2411:
2407:
2400:
2365:
2364:
2357:
2350:
2346:
2342:
2334:
2331:
2326:
2324:
2322:
2321:
2319:
2317:
2310:
2300:
2298:
2293:
2291:
2289:
2282:
2243:
2242:
2235:
2220:
2216:
2181:
2180:
2173:
2108:
2103:
2102:
2095:
2050:
2045:
2044:
2037:
2033:
2023:Dedekind domain
1997:
1993:
1980:
1970:
1959:
1953:
1946:
1944:
1940:
1930:
1912:
1902:
1864:
1863:
1857:
1852:, which is the
1818:
1817:
1772:
1767:
1766:
1758:
1756:
1751:
1740:
1708:
1694:
1693:
1666:
1628:
1627:
1620:integral domain
1618:, which is the
1611:
1604:
1558:
1557:
1493:
1486:
1485:
1471:
1469:
1461:
1447:
1436:
1416:
1414:
1402:
1384:
1380:
1376:
1364:
1362:
1354:
1316:
1315:
1312:
1299:
1258:
1257:
1253:
1233:
1232:
1198:
1173:
1172:
1132:
1118:
1106:
1105:
1098:
1094:
1090:
1080:
1069:
1065:
1010:
1009:
1003:
1000: β 1
996:
952:
951:
944:
940:
905:
904:
900:
865:
828:
827:
823:
815:
813:
799:
797:
796:
792:
754:
753:
751:quadratic field
747:
689:
688:
649:
648:
625:
624:
595:
594:
546:
545:
539:integral domain
503:
502:
488:
482:
476:
462:
461:that satisfies
458:
434:
433:
398:
397:
395:quadratic field
383:
372:
355:
349:
334:
291:
266:
265:
251:
230:Pell's equation
225:
214:
208:
205:
198:needs expansion
183:
169:. The study of
167:quadratic forms
143:
140:
135:
133:
131:
130:
128:
127:
110:
108:
103:
94:
92:
91:
76:
72:
54:
28:
23:
22:
15:
12:
11:
5:
4347:
4345:
4337:
4336:
4331:
4326:
4316:
4315:
4311:
4310:
4300:
4298:
4295:
4293:
4292:
4278:
4260:
4229:
4224:
4219:
4215:
4202:
4200:(3rd ed.)
4193:
4173:
4153:
4118:
4104:
4084:
4082:(2nd ed.)
4074:
4072:
4069:
4066:
4065:
4054:
4043:
4030:
4018:
4006:
4003:de Bruijn 1981
3995:
3983:
3971:
3947:
3935:
3922:
3921:
3919:
3916:
3894:
3893:
3872:
3871:
3844:
3843:
3815:
3809:
3804:
3800:
3793:
3765:
3759:
3755:
3751:
3748:
3743:
3739:
3734:
3730:
3727:
3721:
3716:
3713:
3710:
3707:
3704:
3679:
3676:
3632:
3631:
3607:
3601:
3596:
3592:
3585:
3537:
3534:
3533:
3532:
3500:
3431:β‘ 2,β3 (mod 4)
3378:
3375:
3358:
3352:
3349:
3344:
3341:
3338:
3335:
3332:
3312:
3306:
3303:
3298:
3295:
3292:
3289:
3286:
3250:
3247:
3242:
3239:
3217:
3214:
3209:
3206:
3195:
3194:
3183:
3180:
3175:
3172:
3167:
3164:
3161:
3158:
3153:
3150:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3124:
3121:
3098:
3093:
3087:
3084:
3079:
3075:
3068:
3047:
3046:
3035:
3031:
3024:
3021:
3015:
3010:
3006:
3001:
2995:
2992:
2987:
2983:
2976:
2946:is not even a
2939:
2928:
2915:
2914:
2897:
2892:
2886:
2883:
2878:
2875:
2869:
2864:
2860:
2855:
2849:
2846:
2841:
2837:
2830:
2816:
2796:
2790:
2787:
2782:
2778:
2756:is a complex (
2729:
2726:
2499:
2494:
2491:
2486:
2466:
2463:
2443:
2438:
2435:
2388:
2382:
2377:
2373:
2266:
2260:
2255:
2251:
2204:
2198:
2193:
2189:
2172:
2169:
2135:
2129:
2124:
2120:
2113:
2077:
2071:
2066:
2062:
2055:
1890:
1887:
1881:
1876:
1872:
1841:
1835:
1830:
1826:
1799:
1793:
1788:
1784:
1777:
1725:
1719:
1714:
1711:
1704:
1701:
1665:It is the set
1654:
1651:
1645:
1640:
1636:
1603:
1600:
1581:
1575:
1570:
1566:
1549:
1548:
1537:
1532:
1527:
1524:
1521:
1518:
1513:
1507:
1502:
1499:
1496:
1444:absolute value
1433:
1432:
1373:
1372:
1339:
1333:
1328:
1324:
1311:
1308:
1286:
1283:
1279:
1276:
1271:
1268:
1265:
1250:
1249:
1236:
1230:
1227:
1223:
1220:
1215:
1212:
1209:
1199:
1195:
1189:
1184:
1181:
1175:
1174:
1170:
1167:
1163:
1160:
1155:
1152:
1149:
1146:
1143:
1133:
1129:
1124:
1123:
1121:
1116:
1113:
1101:is defined by
1077:
1076:
1053:
1048:
1041:
1038:
1033:
1028:
1025:
1020:
1017:
993:
992:
981:
976:
971:
968:
965:
962:
959:
928:
922:
917:
913:
886:
883:
877:
872:
868:
862:
858:
854:
851:
845:
840:
836:
780:
777:
771:
766:
762:
746:
743:
720:
715:
712:
707:
702:
699:
696:
674:
669:
664:
659:
656:
634:
610:
605:
602:
572:
569:
563:
558:
554:
526:
520:
515:
511:
442:
421:
415:
410:
406:
322:
318:
314:
309:
306:
303:
298:
294:
288:
285:
282:
279:
276:
273:
263:complex number
250:
247:
216:
215:
195:
193:
182:
179:
101:complex number
69:
68:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4346:
4335:
4332:
4330:
4327:
4325:
4322:
4321:
4319:
4308:
4307:
4302:
4301:
4296:
4289:
4285:
4281:
4275:
4271:
4270:
4265:
4261:
4258:
4254:
4250:
4246:
4242:
4222:
4203:
4199:
4194:
4184:
4183:
4178:
4174:
4170:
4166:
4159:
4154:
4151:on 2015-01-29
4147:
4143:
4139:
4135:
4131:
4124:
4119:
4115:
4111:
4107:
4101:
4097:
4096:Meldrum, John
4093:
4089:
4085:
4081:
4076:
4075:
4070:
4063:
4058:
4055:
4052:
4047:
4044:
4040:
4034:
4031:
4027:
4022:
4019:
4016:, p. 272
4015:
4010:
4007:
4004:
3999:
3996:
3993:, p. 229
3992:
3987:
3984:
3980:
3975:
3972:
3961:
3957:
3951:
3948:
3944:
3943:Bourbaki 1994
3939:
3936:
3932:
3931:Dedekind 1871
3927:
3924:
3917:
3915:
3911:
3906:
3902:
3897:
3889:
3885:
3884:
3883:
3881:
3869:
3864:
3857:
3853:
3852:
3851:
3839:
3835:
3834:
3833:
3807:
3780:
3757:
3753:
3749:
3746:
3741:
3737:
3728:
3719:
3714:
3711:
3708:
3702:
3693:
3691:
3685:
3677:
3675:
3671:
3665:
3663:
3659:
3658:
3653:
3649:
3645:
3641:
3637:
3627:
3623:
3622:
3621:
3599:
3571:
3569:
3565:
3561:
3557:
3553:
3548:
3543:
3535:
3530:
3529:Pierre Fermat
3525:
3517:
3511:
3505:
3501:
3498:
3494:
3485:
3479:
3475:
3448:
3441:
3436:
3435:
3434:
3430:
3425:
3421:
3415:
3411:
3407:
3403:
3393:
3383:
3376:
3374:
3372:
3350:
3347:
3342:
3339:
3336:
3333:
3304:
3301:
3296:
3293:
3290:
3287:
3277:
3272:
3266:
3248:
3245:
3240:
3237:
3215:
3212:
3207:
3204:
3181:
3173:
3170:
3165:
3162:
3151:
3148:
3143:
3140:
3134:
3131:
3128:
3125:
3122:
3119:
3112:
3111:
3110:
3096:
3085:
3082:
3054:
3052:
3033:
3029:
3022:
3019:
3013:
3004:
2993:
2990:
2964:
2963:
2962:
2960:
2956:
2951:
2949:
2945:
2938:
2934:
2927:
2923:
2920:
2912:
2895:
2890:
2884:
2881:
2876:
2873:
2867:
2858:
2847:
2844:
2817:
2814:
2810:
2788:
2785:
2767:
2766:
2765:
2763:
2759:
2742:
2734:
2727:
2725:
2691:
2686:
2679:. For larger
2594:
2541:
2528:
2522:
2518:
2513:
2497:
2489:
2484:
2464:
2461:
2441:
2433:
2419:
2418:
2404:
2380:
2360:
2354:
2313:
2308:
2285:
2280:
2258:
2238:
2232:
2230:
2226:
2196:
2178:
2170:
2168:
2166:
2162:
2158:
2153:
2151:
2127:
2098:
2093:
2069:
2040:
2030:
2028:
2024:
2020:
2016:
2011:
2007:
2004:
2000:
1990:
1987:
1983:
1977:
1974:
1966:
1962:
1956:
1949:
1937:
1933:
1929:
1923:
1919:
1915:
1910:
1905:
1888:
1879:
1860:
1855:
1833:
1815:
1791:
1761:
1754:
1747:
1743:
1723:
1717:
1712:
1709:
1702:
1699:
1689:
1685:
1681:
1677:
1673:
1669:
1652:
1643:
1626:contained in
1625:
1621:
1617:
1609:
1601:
1599:
1597:
1596:
1573:
1555:
1535:
1530:
1525:
1522:
1519:
1516:
1505:
1500:
1497:
1494:
1484:
1483:
1482:
1481:
1474:
1468:
1464:
1458:
1456:
1450:
1445:
1439:
1429:
1425:
1419:
1413:
1409:
1405:
1401:
1400:
1399:
1397:
1393:
1387:
1367:
1361:
1357:
1353:
1352:
1351:
1331:
1309:
1307:
1305:
1281:
1277:
1269:
1266:
1263:
1225:
1221:
1213:
1210:
1207:
1193:
1187:
1182:
1179:
1165:
1161:
1153:
1150:
1147:
1144:
1141:
1127:
1119:
1114:
1111:
1104:
1103:
1102:
1087:
1083:
1075:
1051:
1046:
1039:
1036:
1031:
1026:
1023:
1018:
1015:
1008:
1007:
1006:
999:
979:
974:
969:
966:
963:
960:
957:
950:
949:
948:
920:
897:
884:
875:
870:
866:
852:
843:
818:
811:
805:
802:
778:
769:
752:
744:
742:
740:
736:
718:
713:
705:
700:
697:
672:
667:
662:
657:
654:
632:
608:
603:
600:
592:
589:, the set of
588:
583:
570:
561:
544:
540:
518:
499:
497:
491:
485:
479:
473:
469:
465:
457:
413:
396:
390:
386:
379:
375:
370:
366:
362:
358:
352:
345:
341:
337:
320:
316:
307:
304:
301:
296:
292:
286:
283:
280:
274:
271:
264:
260:
256:
248:
246:
244:
240:
239:
233:
231:
223:
212:
203:
199:
196:This section
194:
191:
187:
186:
180:
178:
176:
172:
168:
164:
160:
155:
153:
126:
125:root of unity
122:
118:
106:
102:
89:
84:
82:
65:
61:
57:
53:
52:
51:
49:
45:
41:
37:
33:
32:number theory
19:
4304:
4303:J.S. Milne.
4268:
4248:
4244:
4241:is Euclidean
4240:
4197:
4187:, retrieved
4181:
4168:
4164:
4146:the original
4133:
4129:
4091:
4079:
4057:
4046:
4038:
4033:
4026:LeVeque 2002
4021:
4009:
3998:
3986:
3974:
3963:. Retrieved
3959:
3950:
3945:, p. 99
3938:
3926:
3909:
3904:
3898:
3895:
3887:
3873:
3855:
3845:
3837:
3694:
3687:
3669:
3666:
3655:
3652:Harold Stark
3648:Kurt Heegner
3633:
3625:
3572:
3564:class number
3546:
3539:
3523:
3515:
3509:
3483:
3474:golden ratio
3446:
3439:
3428:
3413:
3409:
3391:
3388:
3273:
3196:
3055:
3048:
2952:
2943:
2936:
2932:
2925:
2921:
2916:
2747:
2692:= 19, 31, 43
2689:
2685:coefficients
2593:golden ratio
2542:
2526:
2520:
2516:
2511:
2415:
2402:
2358:
2355:
2311:
2283:
2236:
2233:
2174:
2165:OEIS A000924
2161:OEIS A003649
2157:class number
2154:
2149:
2096:
2091:
2038:
2031:
2012:
2005:
2002:
1998:
1988:
1985:
1981:
1978:
1972:
1969:, which has
1964:
1960:
1954:
1947:
1935:
1931:
1928:discriminant
1921:
1917:
1913:
1903:
1858:
1759:
1752:
1745:
1741:
1687:
1683:
1679:
1675:
1671:
1667:
1615:
1605:
1593:
1554:automorphism
1550:
1479:
1472:
1466:
1462:
1459:
1448:
1437:
1434:
1427:
1423:
1417:
1411:
1407:
1403:
1395:
1385:
1374:
1365:
1359:
1355:
1313:
1251:
1085:
1081:
1078:
997:
994:
898:
816:
809:
803:
800:
748:
590:
584:
542:
500:
495:
489:
483:
477:
471:
467:
463:
388:
384:
377:
373:
356:
350:
343:
339:
335:
254:
252:
236:
234:
219:
206:
202:adding to it
197:
170:
156:
104:
88:square roots
85:
80:
70:
63:
59:
55:
35:
29:
4329:Ring theory
4136:: 327β330,
4051:Harper 2004
3636:conjectured
3504:Brahmagupta
3402:real number
3265:irreducible
2909:are called
1952:belongs to
1422: ) =
1388:β‘ 1 (mod 4)
899:An element
541:called the
4318:Categories
4288:1009.11001
4189:2009-08-05
4171:(1): 39β66
4078:Artin, M,
4071:References
4062:Clark 1994
3965:2016-12-31
3860:(sequence
3682:See also:
2813:Carl Gauss
2716:3482 + 531
2706:1520 + 273
1614:defines a
826:) implies
249:Definition
209:March 2015
161:, such as
99:, and the
4266:(2002) .
4251:: 55β70,
3747:−
3371:principal
3357:⟩
3348:−
3343:−
3331:⟨
3311:⟩
3302:−
3285:⟨
3246:−
3241:−
3213:−
3171:−
3166:−
3149:−
3129:⋅
3083:−
3020:−
2991:−
2957:and also
2882:−
2845:−
2786:−
2758:imaginary
2493:¯
2485:−
2462:−
2454:and also
2437:¯
1901:The ring
1700:ω
1598:, below.
1523:−
1512:¯
1480:conjugate
1267:≡
1211:≡
1145:≡
1112:ω
714:⋅
496:imaginary
302:−
287:±
281:−
245:in 1871.
220:Medieval
4334:Integers
4179:(1871),
4090:(1994).
3912:= 14, 69
3850:, when
3570:is one.
3531:in 1657.
3487:, where
3369:are not
3109:we have
2696:170 + 39
2406:, where
2401:Β±
2229:negation
1992:, where
1406: (
1203:if
1137:if
1089:, where
537:form an
494:, it is
365:rational
361:integers
40:integers
4114:1290116
4080:Algebra
3981:, Ch 13
3866:in the
3863:A048981
3472:is the
3469:
3456:√
3451:
2718:√
2708:√
2698:√
2673:√
2663:√
2656:
2643:√
2638:
2629:√
2619:√
2609:√
2599:√
2588:
2575:√
2570:
2561:√
2551:√
2532:, with
2524:√
2338:
2325:√
2320:
2299:√
2292:√
2290:1, β1,
1945:√
1757:√
1470:√
1415:√
1390:, both
1363:√
995:or, if
814:√
812:
798:√
348:, with
181:History
147:
134:√
129:
109:√
93:√
4286:
4276:
4112:
4102:
3832:when
3672:> 0
3644:proven
3394:> 0
3276:ideals
2931:) and
2807:, the
2683:, the
2661:15 + 4
2627:10 + 3
2361:> 0
2239:< 0
2099:< 0
2041:> 0
2019:ideals
1926:whose
1750:, and
1692:where
1606:Every
1592:β see
1478:has a
1451:> 0
1440:< 0
1394:. The
1375:where
1304:square
791:where
741:four.
739:degree
492:< 0
391:< 0
380:> 0
257:is an
123:cubic
4161:(PDF)
4149:(PDF)
4126:(PDF)
3979:Artin
3918:Notes
3777:as a
3640:Gauss
3620:for
3424:units
2607:8 + 3
2597:5 + 2
2591:(the
2323:Β±1 Β±
2171:Units
2094:. If
2036:. If
1909:roots
1072:both
1064:with
737:have
481:. If
132:β1 +
71:with
4274:ISBN
4100:ISBN
3905:real
3868:OEIS
3642:and
3540:The
3513:β 61
3437:For
3418:, a
3389:For
3323:and
3274:The
3263:are
3230:and
2748:For
2714:and
2671:4 +
2641:3 +
2617:3 +
2573:1 +
2559:2 +
2549:1 +
2547:are
2536:and
2477:and
2349:and
2318:Β±1,
2314:= β3
2286:= β1
2177:unit
2013:The
1396:norm
1379:and
1306:4).
1252:(as
1093:and
1068:and
943:and
687:and
623:and
587:ring
354:and
121:real
75:and
4284:Zbl
4253:doi
4243:",
4138:doi
3646:by
3638:by
3519:= 1
3442:= 5
3433:.
3416:= 1
3056:In
2595:),
2512:the
2356:If
2297:, β
2234:If
2219:or
2152:.
2008:= 0
1967:= 0
1934:β 4
1924:= 0
1862:in
1856:of
1816:of
1748:+ 1
1744:= 4
1739:if
1670:= {
1278:mod
1222:mod
1162:mod
1074:odd
903:of
591:all
466:β 4
387:β 4
376:β 4
371:if
346:= 0
204:.
66:= 0
42:to
30:In
4320::
4282:.
4249:56
4247:,
4223:14
4169:43
4167:,
4163:,
4134:83
4132:,
4128:,
4110:MR
4108:.
3958:.
3870:).
3781:,
3664:.
3454:1+
3449:=
3444:,
3414:DY
3412:β
3396:,
3269:Β±1
2950:.
2764:.
2724:.
2720:43
2710:31
2704:,
2700:19
2675:15
2669:,
2665:14
2659:,
2645:13
2635:,
2631:11
2625:,
2621:10
2615:,
2605:,
2567:,
2557:,
2519:+
2353:.
2351:β1
2327:β3
2301:β1
2294:β1
2221:β1
2167:.
2029:.
2010:.
2001:β
1984:=
1963:β
1920:+
1918:Bx
1916:+
1755:=
1690:},
1686:β
1682:,
1678::
1676:Οb
1674:+
1465:+
1428:Db
1426:β
1410:+
1358:+
1086:Οb
1084:+
808:=
472:De
470:=
342:+
340:bx
338:+
253:A
232:.
177:.
154:.
136:β3
111:β1
107:=
83:.
62:+
60:bx
58:+
34:,
4290:.
4255::
4228:]
4218:[
4214:Z
4140::
4116:.
3968:.
3910:D
3892:,
3888:D
3876:D
3856:D
3848:D
3842:,
3838:D
3830:D
3814:)
3808:D
3803:(
3799:Q
3792:O
3764:|
3758:2
3754:b
3750:D
3742:2
3738:a
3733:|
3729:=
3726:)
3720:D
3715:b
3712:+
3709:a
3706:(
3703:N
3670:D
3630:.
3626:D
3606:)
3600:D
3595:(
3591:Q
3584:O
3547:Z
3524:Z
3516:Y
3510:X
3499:.
3489:n
3484:Ο
3482:Β±
3466:2
3463:/
3458:5
3447:Ο
3440:D
3429:D
3410:X
3398:Ο
3392:D
3351:5
3340:1
3337:,
3334:3
3305:5
3297:+
3294:1
3291:,
3288:3
3249:5
3238:2
3216:5
3208:+
3205:2
3182:.
3179:)
3174:5
3163:2
3160:(
3157:)
3152:5
3144:+
3141:2
3138:(
3135:=
3132:3
3126:3
3123:=
3120:9
3097:,
3092:)
3086:5
3078:(
3074:Q
3067:O
3034:,
3030:]
3023:5
3014:[
3009:Z
3005:=
3000:)
2994:5
2986:(
2982:Q
2975:O
2944:Z
2940:3
2937:ΞΆ
2935:(
2933:Q
2929:4
2926:ΞΆ
2924:(
2922:Q
2913:.
2896:]
2891:2
2885:3
2877:+
2874:1
2868:[
2863:Z
2859:=
2854:)
2848:3
2840:(
2836:Q
2829:O
2795:]
2789:1
2781:[
2777:Z
2754:Ο
2750:D
2690:D
2681:D
2653:2
2650:/
2611:7
2601:6
2585:2
2582:/
2577:5
2563:3
2553:2
2545:D
2538:b
2534:a
2527:D
2521:b
2517:a
2498:.
2490:u
2465:u
2442:,
2434:u
2422:u
2412:u
2408:i
2403:u
2387:)
2381:D
2376:(
2372:Q
2359:D
2347:1
2343:D
2335:2
2332:/
2312:D
2309:(
2284:D
2281:(
2265:)
2259:D
2254:(
2250:Q
2237:D
2217:1
2203:)
2197:D
2192:(
2188:Q
2134:)
2128:D
2123:(
2119:Q
2112:O
2097:D
2076:)
2070:D
2065:(
2061:Q
2054:O
2039:D
2034:D
2006:D
2003:m
1999:x
1994:D
1989:D
1986:m
1982:n
1973:D
1971:4
1965:D
1961:x
1955:Z
1948:D
1941:D
1936:C
1932:B
1922:C
1914:x
1904:Z
1889:.
1886:)
1880:D
1875:(
1871:Q
1859:Z
1840:)
1834:D
1829:(
1825:Q
1798:)
1792:D
1787:(
1783:Q
1776:O
1760:D
1753:Ο
1746:k
1742:D
1724:2
1718:D
1713:+
1710:1
1703:=
1688:Z
1684:b
1680:a
1672:a
1668:Z
1653:.
1650:)
1644:D
1639:(
1635:Q
1612:D
1580:)
1574:D
1569:(
1565:Q
1536:.
1531:D
1526:b
1520:a
1517:=
1506:D
1501:b
1498:+
1495:a
1473:D
1467:b
1463:a
1449:D
1438:D
1431:.
1424:a
1418:D
1412:b
1408:a
1404:N
1386:D
1381:b
1377:a
1371:,
1366:D
1360:b
1356:a
1338:)
1332:D
1327:(
1323:Q
1300:D
1285:)
1282:4
1275:(
1270:0
1264:D
1254:D
1229:)
1226:4
1219:(
1214:1
1208:D
1194:2
1188:D
1183:+
1180:1
1169:)
1166:4
1159:(
1154:3
1151:,
1148:2
1142:D
1128:D
1120:{
1115:=
1099:Ο
1095:b
1091:a
1082:a
1070:b
1066:a
1052:,
1047:D
1040:2
1037:b
1032:+
1027:2
1024:a
1019:=
1016:x
1004:4
998:D
980:,
975:D
970:b
967:+
964:a
961:=
958:x
945:b
941:a
927:)
921:D
916:(
912:Q
901:x
885:.
882:)
876:D
871:2
867:a
861:(
857:Q
853:=
850:)
844:D
839:(
835:Q
824:a
817:D
810:a
804:D
801:a
793:D
779:,
776:)
770:D
765:(
761:Q
719:3
711:)
706:2
701:+
698:1
695:(
673:3
668:+
663:2
658:+
655:1
633:3
609:2
604:+
601:1
571:.
568:)
562:D
557:(
553:Q
525:)
519:D
514:(
510:Q
490:D
484:D
478:e
468:c
464:b
459:D
441:Q
420:)
414:D
409:(
405:Q
389:c
385:b
378:c
374:b
357:c
351:b
344:c
336:x
321:2
317:/
313:)
308:c
305:4
297:2
293:b
284:b
278:(
275:=
272:x
226:D
211:)
207:(
144:2
141:/
105:i
95:2
77:c
73:b
64:c
56:x
20:)
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