95:
85:
64:
31:
1214:. Davenport (p.39) says "The relation (2) between an imprimitive character and the primitive character which induces it implies a simple relation between the corresponding L-functions" and follows with a formula involving the Euler factors at primes dividing the modulus of the imprimitive factor. Apostol (p.262) says "every L-series is equal to the L-series of a primitive character, multiplied by a finite number of factors". Suggestions for a simpler summary would be welcome.
3341:
22:
985:), this should be written explicitly, but even then it is confusing as one usually speaks about periods for functions defined on integers, not functions defined on finite groups, and the definition of period in such a case would require an explanation. The quoted description by Davenport is yet different: he considers the character as a function on the subset
2932:
1179:), so no Euler factor is missing. This is not a problem if we define characters purely as completely multiplicative periodic functions on the integers, since then χ and ψ are identical characters. However, if we either consider a specification of a modulus as a part of the definition of a character, or consider characters on (
1276:
primitive character, but the former is then incorrect as a character can have smaller induced moduli without being induced by another character. For example, the unique non-trivial character modulo 6 has induced modulus 3 but is not an extension of any character mod 1,2, or 3. Do people agree with me here?
894:". I think that what I wrote is a fair rendition of that, and if not I invite improvements. However, EmilJ's comment is wrong in itself. The principal character has as its associated character the multiplicative function which is always 1, and this has period 1. So by my rendition it is not primitive.
481:
Just wondering something... of the two mod 4 characters on this page, the first {1,0,1,0} seems to be the same as the sole mod 2 character {1,0}. Does it really need to be in the list twice? I think it should be deleted from the mod 4 list, considering its period is 2; that would leave the sole mod 4
2250:
if prime) and how we prove that the
Dirichlet character of a same periodicity are all orthogonal. Because two integers always have a lcm, two Dirichlet characters can always thought as being of the same periodicity, and thus are mutually orthogonal. So that all the Dirichlet characters together form
494:
I don't think it would be correct to remove it, as that would make it appear as if there were really only one character mod 4. There is a theorem about the number of characters mod n, and it tells us that there are 2 mod 4 characters - it just happens that one of them is the same as the only mod 2
1237:
divides the modulus but not the conductor, but I don’t think it’s necessary to spell this out, it’s kind of obvious from the Euler factor expansion itself.) Also, characters—primitive or imprimitive—“correspond” to L-functions, if anything, not to missing factors, making the new wording clumsy. I
2927:
1275:
Currently there are two statements defining a primitive character: "A character is primitive if it is not induced by any character of smaller modulus" and "A character is primitive if there is no smaller induced modulus". But these aren't the same, the latter is the now common definition of a
1140:
associated with the principal
Dirichlet character is the character which always takes the value 1. This is defined on a smaller modulus, namely 1. (As it happens, Davenport at p.37 thinks it "a matter of personal preference whether one includes the principal character among the imprimitive
3336:{\displaystyle L(\chi ,s)=\sum _{n=1}^{\infty }\chi (n)n^{-s}=\int _{0}^{\infty }\sum _{n}\chi (n)\delta (x-n)x^{-s}dx=\sum _{k=1}^{\infty }{\bar {\chi }}(k)k^{s-1}{\frac {1}{q}}\int _{0}^{\infty }\left(Y(1)e^{-2i\pi kx/q}+Y(-1)e^{2i\pi kx/q}-2\right)x^{-s}dx=L(1-s,\chi )A(s)}
455:
I am not an expert on
Dirichlet characters, but ... are we sure that the first and third conditions really are sufficient? - Doesn't the identically zero function satisfy the first and third conditions, whilst not being a Dirichlet character?
8846:
2412:
2647:
777:
2717:
2154:
The same idea can be extended to characters of non-prime periodicity. After that, we remark that a change of generator can be seen as a change of root of the unity, so that all the
Dirichlet characters of periodicity
8722:
8669:
8254:
7839:
8775:
8478:
8030:
7745:
7651:
7460:
7179:
7051:
6792:
6543:
6090:
4116:
3907:
437:
If it is clutter, maybe it'd be better to show only the table for 7 - I think the rest are pretty obvious once you understand those. I've corrected the comment at the top to promise only what is actually there.
5029:
2502:
1228:
The original wording is in fact fine. The “can” takes care of the problem that not every imprimitive characters has factors missing in its L-function. (The exact characterization is that the factor for
1856:
1444:
859:
An equivalent definition of a primitive
Dirichlet character is to consider the associated multiplicative character and define it to be primitive if the period of the character is exactly the modulus.
4919:
8974:
1642:
1503:
8384:
8160:
6423:
5870:
4480:
5204:
4862:
4400:
4007:
3965:
2304:
3669:
5373:
4812:
4738:
4598:
1041:
151:
8888:
8616:
8520:
8425:
8304:
8201:
8080:
7977:
7881:
7786:
7692:
7598:
7502:
7407:
7311:
7126:
6998:
6862:
6739:
6609:
6490:
4777:
4703:
4668:
4633:
4563:
4063:
3854:
6343:
6272:
6146:
6037:
5925:
5790:
5671:
5567:
5519:
5250:
5096:
4962:
4354:
4319:
4284:
4249:
4210:
4175:
3786:
3751:
3704:
3552:
3509:
3470:
3416:
2712:
1371:
1990:
1944:
829:
9131:
2148:
629:
557:
2032:
1800:
1098:
8943:
8575:
7936:
7557:
7366:
6917:
6664:
1719:
6201:
5980:
5726:
5622:
3607:
2077:
5323:
1163:
This is sensitive to the choice of what kind of characters one considers. For example, let χ be the nonprincipal character mod 4, and ψ be the induced character mod 8. Then
399:
286:
5158:
5125:
2202:
1756:
337:
227:
866:
1 has period n, but it is not primitive." I assume it's me rather than
Davenport who was wrong here? Anyway, the source says "It is possible however that for values of
664:
5049:
5413:
9121:
8780:
2522:
580:
2228:
1882:
9136:
5393:
2248:
2173:
2097:
1669:
1583:
1563:
1543:
1523:
1391:
5290:
909:
I think the terminology you've used is non-standard and has lead to confusion. By "the associated multiplicative character", I assume you mean the character of (
2309:
2527:
35:
424:
Yes, that is correct. But perhaps these can continue to be left out, for, I fear, they would contribute to a mind-numbing clutter? Sometimes, less is more.
9146:
141:
8996:
I completely rewrote this article. Unfortunately, I submitted it for review as though it were a new article, so it's not on my sandbox page but is here:
669:
187:
It seems obvious to me that the following tables are missing, but since I know nothing of this subject beyond what I've just read here I better check:
9116:
2922:{\displaystyle \sum _{n}\chi (n)\delta (x-n)={\frac {1}{q}}\sum _{k=1}^{\infty }{\bar {\chi }}(k)\left(Y(1)e^{-2i\pi kx/q}+Y(-1)e^{2i\pi kx/q}\right)}
1304:
I don’t get your point. The nontrivial character mod 6 is induced by the nontrivial character mod 3, so it is imprimitive under either definition. If
2175:
can be constructed from the same generator but with different roots of the unity. And this is how we get to the discrete
Fourier transform (of size
9126:
1312:
such that χ is induced by ψ, so the two definitions are equivalent (they are pretty much just saying the same thing in slightly different words).—
940:
consider the associated character on multiplicative groups and define it to be primitive if the character is not defined on any smaller modulus
117:
974:), and 0 otherwise. This function does not have period 1. As RobHar wrote, your terminology is confusing. If you really meant a character of (
9141:
8674:
8621:
8206:
7791:
8727:
8430:
7982:
7697:
7603:
7412:
7131:
7003:
6744:
6495:
6042:
4068:
3859:
1108:. Note the indefinite article: a periodic function does not have a unique period, any multiple will do, and in particular, a character mod
8967:
Riemann might have extended the original definition of the Zeta function. I am not sure that he referred to all the L(s,chi) functions.
3362:
838:
831:. This makes me wonder how many other mistakes are in the tables for the Dirichlet characters; these should all be carefully checked.
8978:
1292:
108:
69:
9111:
4983:
2417:
1808:
1396:
4867:
1116:. Only the minimal period (= conductor) of an imprimitive character (in Davenport’s sense) is guaranteed to be less than
8993:
I used to edit math articles under the name
Virginia-American but my computer died, etc., and I just recently got back.
3358:
complex analysis, Fourier analysis, Dirichlet series, Mellin/Laplace/Fourier transforms, zeta functions / L-functions...
44:
1588:
1449:
8309:
8085:
6348:
5795:
4405:
865:
deleted that with the edit summary "This is wrong. For example, the principal character of square-free modulus n : -->
5167:
4825:
4363:
3970:
3928:
2254:
3628:
5328:
4782:
4708:
4568:
988:
8997:
8858:
8586:
8490:
8395:
8274:
8171:
8050:
7947:
7851:
7756:
7662:
7568:
7472:
7377:
7281:
7096:
6968:
6832:
6709:
6579:
6460:
4747:
4673:
4638:
4603:
4533:
4028:
3819:
6313:
6242:
6116:
6007:
5895:
5760:
5641:
5537:
5489:
5220:
5066:
4932:
4324:
4289:
4254:
4219:
4180:
4145:
3756:
3721:
3674:
3522:
3479:
3440:
3386:
2652:
1341:
1238:
would go with the original wording, maybe slightly reformulated so as not to make characters causative agents:
1951:
1887:
782:
2102:
842:
3366:
920:). Then, you are correct, but EmilJ probably thought you meant the multiplicative function on the integers.
1288:
585:
513:
1995:
1761:
1050:
9093:
9063:
9039:
9015:
8893:
8525:
7886:
7507:
7316:
6867:
6614:
1674:
1259:
1219:
1146:
947:
899:
414:
Since the text claims to present "all of the characters up to modulus 7", these should be included, no?
50:
6151:
5930:
5676:
5572:
3557:
94:
1284:
8970:
1280:
834:
485:
1190:), then the two characters are distinct, and ψ is imprimitive. Presumably this should be clarified.—
21:
5052:
2037:
5295:
116:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
8841:{\displaystyle \left({\frac {a}{2}}\right)\left({\frac {a}{3}}\right)\left({\frac {a}{5}}\right)}
368:
255:
100:
5134:
5101:
2178:
84:
63:
1726:
938:
Good point, perhaps "character on the multiplicative group" would have been better. How about
307:
197:
642:
5034:
1161:
The imprimitive characters correspond to missing Euler factors in the associated L-functions.
9089:
9059:
9035:
9011:
6237:
5398:
1255:
1215:
1212:
The imprimitive characters correspond to missing Euler factors in the associated L-functions
1142:
1141:
characters, I prefer to leave it unclassified".) I already suggested an improved wording.
943:
925:
895:
9053:
My rewrite has been posted for a month. No comments, corrections, or anything from onyone
2507:
2407:{\displaystyle \sum _{n}{\frac {\chi (n)}{n^{s}}}=\prod _{p}{\frac {1}{1-\chi (p)p^{-s}}}}
1316:
1246:
1194:
1124:
562:
496:
457:
3345:
Dirichlet characters are thus a nice mathematical construction based on no less than :
2642:{\displaystyle \forall a\in G,\quad \sum _{n\in G}\chi (an)e^{2i\pi ank/p}=\chi (a)Y(k)}
2207:
1861:
9003:
I think my rewrite is a big improvement, but I'd like some feedback before I post it.
5378:
2233:
2158:
2082:
1654:
1568:
1548:
1528:
1508:
1376:
9105:
5263:
5128:
772:{\displaystyle \chi _{2}(2)=\chi _{2}(3)^{2}=(\omega ^{2})^{2}=\omega ^{4}=-\omega }
631:
9080:
If anyone has anything constructive (errors, additions, whatever) to say, say it.
1240:
Imprimitivity of characters can lead to missing Euler factors in their L-functions
1100:
whenever both sides are defined. Then a character is indeed imprimitive if it has
921:
113:
861:
and cited
Davenport (1967). I didn't give the page number, but it's page 37.
9097:
9067:
9043:
9019:
8982:
3370:
1319:
1313:
1296:
1263:
1249:
1243:
1223:
1197:
1191:
1150:
1127:
1121:
951:
929:
903:
862:
846:
634:
499:
488:
470:
467:
460:
442:
428:
425:
418:
177:
174:
90:
173:
Historically, what does the L in L-function or L-series stand for? Lagrange?
9083:
And, could someone more knowledgeable than me post categories and rankings
5749:= 4 (4 = 2, the smallest primitive root mod 2 is 1, and 3 is the smallest
1324:
I was confusing something with induced and extending characters. My bad.
8717:{\displaystyle \left({\frac {a}{3}}\right)\left({\frac {a}{10}}\right)}
8664:{\displaystyle \left({\frac {a}{2}}\right)\left({\frac {a}{15}}\right)}
8249:{\displaystyle \left({\frac {a}{2}}\right)\left({\frac {a}{13}}\right)}
7834:{\displaystyle \left({\frac {a}{2}}\right)\left({\frac {a}{11}}\right)}
510:
There was an error in this table which I have corrected. The error set
439:
415:
8770:{\displaystyle \left({\frac {a}{5}}\right)\left({\frac {a}{6}}\right)}
8473:{\displaystyle \left({\frac {a}{4}}\right)\left({\frac {a}{7}}\right)}
8263:
27 = 3, the smallest primitive root mod 3 is 2, and 2 is the smallest
8039:
25 = 5, the smallest primitive root mod 5 is 2, and 2 is the smallest
8025:{\displaystyle \left({\frac {a}{3}}\right)\left({\frac {a}{8}}\right)}
7740:{\displaystyle \left({\frac {a}{3}}\right)\left({\frac {a}{7}}\right)}
7646:{\displaystyle \left({\frac {a}{4}}\right)\left({\frac {a}{5}}\right)}
7455:{\displaystyle \left({\frac {a}{2}}\right)\left({\frac {a}{9}}\right)}
7174:{\displaystyle \left({\frac {a}{3}}\right)\left({\frac {a}{5}}\right)}
7046:{\displaystyle \left({\frac {a}{2}}\right)\left({\frac {a}{7}}\right)}
6787:{\displaystyle \left({\frac {a}{3}}\right)\left({\frac {a}{4}}\right)}
6538:{\displaystyle \left({\frac {a}{2}}\right)\left({\frac {a}{5}}\right)}
6085:{\displaystyle \left({\frac {a}{2}}\right)\left({\frac {a}{3}}\right)}
4111:{\displaystyle \left({\frac {a}{m}}\right)\left({\frac {a}{n}}\right)}
3902:{\displaystyle \left({\frac {a}{n}}\right)\left({\frac {b}{n}}\right)}
6302:
9 = 3, the smallest primitive root mod 3 is 2, and 2 is the smallest
1208:
Imprimitive characters can cause missing Euler factors in L-functions
8271:≡ 2 (mod 3) which is also a primitive root mod 27, thus we define
8047:≡ 2 (mod 5) which is also a primitive root mod 25, thus we define
6310:≡ 2 (mod 3) which is also a primitive root mod 9, thus we define
5757:≡ 1 (mod 2) which is also a primitive root mod 4, thus we define
2251:
an orthogonal basis of the vector space of periodic sequences
1308:
is an induced modulus of χ, there is indeed a character ψ mod
15:
3380:(This character is a special example of Dirichlet character)
2714:. from that we can write a Fourier series representation of
5024:{\displaystyle \left({\frac {a}{n}}\right)^{\lambda (n)}}
2497:{\displaystyle Y(k)=\sum _{n\in G}\chi (n)e^{2i\pi nk/p}}
1851:{\displaystyle G=(\mathbb {Z} /p\mathbb {Z} ,\times )}
1439:{\displaystyle G=(\mathbb {Z} /p\mathbb {Z} ,\times )}
1043:
of integers, which is presumably meant to have period
8896:
8861:
8783:
8730:
8677:
8624:
8589:
8528:
8493:
8433:
8398:
8312:
8277:
8209:
8174:
8088:
8053:
7985:
7950:
7889:
7854:
7794:
7759:
7700:
7665:
7606:
7571:
7510:
7475:
7415:
7380:
7319:
7284:
7134:
7099:
7006:
6971:
6870:
6835:
6747:
6712:
6617:
6582:
6498:
6463:
6351:
6316:
6245:
6154:
6119:
6045:
6010:
5933:
5898:
5798:
5763:
5679:
5644:
5575:
5540:
5492:
5401:
5381:
5331:
5298:
5266:
5223:
5170:
5137:
5104:
5069:
5037:
4986:
4935:
4914:{\displaystyle \left({\frac {2^{k}-a}{2^{k}}}\right)}
4870:
4828:
4785:
4750:
4711:
4676:
4641:
4606:
4571:
4536:
4408:
4366:
4327:
4292:
4257:
4222:
4183:
4148:
4071:
4031:
3973:
3931:
3862:
3822:
3759:
3724:
3677:
3631:
3560:
3525:
3482:
3443:
3389:
2935:
2720:
2655:
2530:
2510:
2420:
2312:
2257:
2236:
2210:
2181:
2161:
2105:
2085:
2040:
1998:
1954:
1890:
1864:
1811:
1764:
1729:
1677:
1657:
1591:
1571:
1551:
1531:
1511:
1452:
1399:
1379:
1344:
1053:
991:
785:
672:
645:
588:
565:
516:
371:
310:
258:
200:
8855:
smallest primitive root mod 31 is 3, thus we define
8487:
smallest primitive root mod 29 is 2, thus we define
7848:
smallest primitive root mod 23 is 5, thus we define
7469:
smallest primitive root mod 19 is 2, thus we define
7278:
smallest primitive root mod 17 is 3, thus we define
6829:
smallest primitive root mod 13 is 2, thus we define
6576:
smallest primitive root mod 11 is 2, thus we define
639:
There is actually a mistake in this table. Consider
112:, a collaborative effort to improve the coverage of
6113:smallest primitive root mod 7 is 3, thus we define
5892:smallest primitive root mod 5 is 2, thus we define
5638:smallest primitive root mod 3 is 2, thus we define
5534:smallest primitive root mod 2 is 1, thus we define
1335:This is how I would try to introduce the concept :
8937:
8882:
8840:
8769:
8716:
8663:
8610:
8569:
8514:
8472:
8419:
8378:
8298:
8248:
8195:
8154:
8074:
8024:
7971:
7930:
7875:
7833:
7780:
7739:
7686:
7645:
7592:
7551:
7496:
7454:
7401:
7360:
7305:
7173:
7120:
7045:
6992:
6911:
6856:
6786:
6733:
6658:
6603:
6537:
6484:
6417:
6337:
6266:
6195:
6140:
6084:
6031:
5974:
5919:
5864:
5784:
5720:
5665:
5616:
5561:
5513:
5407:
5387:
5367:
5317:
5284:
5244:
5198:
5152:
5119:
5090:
5043:
5023:
4956:
4913:
4856:
4806:
4771:
4732:
4697:
4662:
4627:
4592:
4557:
4474:
4394:
4348:
4313:
4278:
4243:
4204:
4169:
4110:
4057:
4001:
3959:
3901:
3848:
3780:
3745:
3698:
3663:
3601:
3546:
3503:
3464:
3410:
3335:
2921:
2706:
2641:
2516:
2496:
2406:
2298:
2242:
2222:
2196:
2167:
2142:
2091:
2071:
2026:
1984:
1938:
1876:
1850:
1794:
1750:
1713:
1663:
1637:{\displaystyle (\mathbb {Z} /(p-1)\mathbb {Z} ,+)}
1636:
1577:
1557:
1537:
1517:
1498:{\displaystyle (\mathbb {Z} /(p-1)\mathbb {Z} ,+)}
1497:
1438:
1385:
1365:
1092:
1035:
823:
771:
658:
623:
574:
551:
466:It appears someone cleaned up the article already
393:
331:
280:
221:
8379:{\displaystyle e^{{2\pi i}/((3-1)\cdot 3^{3-1})}}
8155:{\displaystyle e^{{2\pi i}/((5-1)\cdot 5^{2-1})}}
6418:{\displaystyle e^{{2\pi i}/((3-1)\cdot 3^{2-1})}}
5865:{\displaystyle e^{{2\pi i}/((2-1)\cdot 2^{2-1})}}
4522:, but not always, the smallest counterexample is
4475:{\displaystyle e^{{2\pi i}/((p-1)\cdot p^{k-1})}}
344:
236:
9132:Knowledge level-5 vital articles in Mathematics
5199:{\displaystyle \left({\frac {a}{n}}\right)^{k}}
4857:{\displaystyle \left({\frac {a}{2^{k}}}\right)}
4395:{\displaystyle \left({\frac {a}{p^{k}}}\right)}
4002:{\displaystyle \left({\frac {a}{n}}\right)^{k}}
3960:{\displaystyle \left({\frac {a^{k}}{n}}\right)}
2299:{\displaystyle (u_{n})_{n\in \mathbb {N} ^{*}}}
1671:which will be the periodicity of the character
3664:{\displaystyle \left({\frac {a+kn}{n}}\right)}
1647:The Dirichlet character exploits this idea :
962:is the multiplicative function which is 1 for
5368:{\displaystyle r(\cos \theta +i\sin \theta )}
4807:{\displaystyle \left({\frac {11}{16}}\right)}
4733:{\displaystyle \left({\frac {13}{16}}\right)}
4593:{\displaystyle \left({\frac {15}{16}}\right)}
1036:{\displaystyle \{q\in \mathbb {Z} :(n,q)=1\}}
8:
8883:{\displaystyle \left({\frac {3}{31}}\right)}
8611:{\displaystyle \left({\frac {a}{30}}\right)}
8515:{\displaystyle \left({\frac {2}{29}}\right)}
8420:{\displaystyle \left({\frac {a}{28}}\right)}
8299:{\displaystyle \left({\frac {2}{27}}\right)}
8196:{\displaystyle \left({\frac {a}{26}}\right)}
8075:{\displaystyle \left({\frac {2}{25}}\right)}
7972:{\displaystyle \left({\frac {a}{24}}\right)}
7876:{\displaystyle \left({\frac {5}{23}}\right)}
7781:{\displaystyle \left({\frac {a}{22}}\right)}
7687:{\displaystyle \left({\frac {a}{21}}\right)}
7593:{\displaystyle \left({\frac {a}{20}}\right)}
7497:{\displaystyle \left({\frac {2}{19}}\right)}
7402:{\displaystyle \left({\frac {a}{18}}\right)}
7306:{\displaystyle \left({\frac {3}{17}}\right)}
7121:{\displaystyle \left({\frac {a}{15}}\right)}
6993:{\displaystyle \left({\frac {a}{14}}\right)}
6857:{\displaystyle \left({\frac {2}{13}}\right)}
6734:{\displaystyle \left({\frac {a}{12}}\right)}
6604:{\displaystyle \left({\frac {2}{11}}\right)}
6485:{\displaystyle \left({\frac {a}{10}}\right)}
4772:{\displaystyle \left({\frac {5}{16}}\right)}
4698:{\displaystyle \left({\frac {3}{16}}\right)}
4663:{\displaystyle \left({\frac {9}{16}}\right)}
4628:{\displaystyle \left({\frac {7}{16}}\right)}
4558:{\displaystyle \left({\frac {1}{16}}\right)}
4058:{\displaystyle \left({\frac {a}{mn}}\right)}
3849:{\displaystyle \left({\frac {ab}{n}}\right)}
1030:
992:
9077:Two weeks later, I'm reposting my rewrite.
6338:{\displaystyle \left({\frac {2}{9}}\right)}
6267:{\displaystyle \left({\frac {a}{8}}\right)}
6141:{\displaystyle \left({\frac {3}{7}}\right)}
6032:{\displaystyle \left({\frac {a}{6}}\right)}
5920:{\displaystyle \left({\frac {2}{5}}\right)}
5785:{\displaystyle \left({\frac {3}{4}}\right)}
5666:{\displaystyle \left({\frac {2}{3}}\right)}
5562:{\displaystyle \left({\frac {1}{2}}\right)}
5514:{\displaystyle \left({\frac {0}{1}}\right)}
5245:{\displaystyle \left({\frac {a}{n}}\right)}
5091:{\displaystyle \left({\frac {a}{n}}\right)}
4957:{\displaystyle \left({\frac {a}{n}}\right)}
4518:is usually the smallest primitive root mod
4349:{\displaystyle \left({\frac {5}{8}}\right)}
4314:{\displaystyle \left({\frac {3}{8}}\right)}
4279:{\displaystyle \left({\frac {7}{8}}\right)}
4244:{\displaystyle \left({\frac {1}{8}}\right)}
4205:{\displaystyle \left({\frac {3}{4}}\right)}
4170:{\displaystyle \left({\frac {1}{4}}\right)}
3781:{\displaystyle \left({\frac {b}{n}}\right)}
3746:{\displaystyle \left({\frac {a}{n}}\right)}
3699:{\displaystyle \left({\frac {a}{n}}\right)}
3617:is the smallest positive primitive root of
3547:{\displaystyle \left({\frac {a}{p}}\right)}
3504:{\displaystyle \left({\frac {1}{n}}\right)}
3465:{\displaystyle \left({\frac {0}{1}}\right)}
3411:{\displaystyle \left({\frac {a}{n}}\right)}
8968:
2707:{\displaystyle Y(k)=Y(1){\bar {\chi }}(n)}
1366:{\displaystyle \mathbb {Z} /p\mathbb {Z} }
1159:I also have a quibble about the statement
1132:The character on the multiplicative group
832:
58:
8913:
8902:
8901:
8895:
8866:
8860:
8824:
8806:
8788:
8782:
8753:
8735:
8729:
8700:
8682:
8676:
8647:
8629:
8623:
8594:
8588:
8545:
8534:
8533:
8527:
8498:
8492:
8456:
8438:
8432:
8403:
8397:
8359:
8329:
8318:
8317:
8311:
8282:
8276:
8232:
8214:
8208:
8179:
8173:
8135:
8105:
8094:
8093:
8087:
8058:
8052:
8008:
7990:
7984:
7955:
7949:
7906:
7895:
7894:
7888:
7859:
7853:
7817:
7799:
7793:
7764:
7758:
7723:
7705:
7699:
7670:
7664:
7629:
7611:
7605:
7576:
7570:
7527:
7516:
7515:
7509:
7480:
7474:
7438:
7420:
7414:
7385:
7379:
7336:
7325:
7324:
7318:
7289:
7283:
7157:
7139:
7133:
7104:
7098:
7029:
7011:
7005:
6976:
6970:
6887:
6876:
6875:
6869:
6840:
6834:
6770:
6752:
6746:
6717:
6711:
6634:
6623:
6622:
6616:
6587:
6581:
6521:
6503:
6497:
6468:
6462:
6398:
6368:
6357:
6356:
6350:
6321:
6315:
6250:
6244:
6171:
6160:
6159:
6153:
6124:
6118:
6068:
6050:
6044:
6015:
6009:
5950:
5939:
5938:
5932:
5903:
5897:
5845:
5815:
5804:
5803:
5797:
5768:
5762:
5696:
5685:
5684:
5678:
5649:
5643:
5592:
5581:
5580:
5574:
5545:
5539:
5497:
5491:
5400:
5380:
5330:
5306:
5297:
5265:
5228:
5222:
5190:
5176:
5169:
5136:
5103:
5074:
5068:
5036:
5006:
4992:
4985:
4940:
4934:
4899:
4882:
4875:
4869:
4842:
4833:
4827:
4790:
4784:
4755:
4749:
4716:
4710:
4681:
4675:
4646:
4640:
4611:
4605:
4576:
4570:
4541:
4535:
4455:
4425:
4414:
4413:
4407:
4380:
4371:
4365:
4332:
4326:
4297:
4291:
4262:
4256:
4227:
4221:
4188:
4182:
4153:
4147:
4094:
4076:
4070:
4036:
4030:
3993:
3979:
3972:
3942:
3936:
3930:
3885:
3867:
3861:
3827:
3821:
3764:
3758:
3729:
3723:
3682:
3676:
3636:
3630:
3577:
3566:
3565:
3559:
3530:
3524:
3487:
3481:
3448:
3442:
3394:
3388:
3279:
3254:
3238:
3206:
3187:
3160:
3155:
3141:
3129:
3105:
3104:
3098:
3087:
3065:
3025:
3015:
3010:
2994:
2972:
2961:
2934:
2904:
2888:
2856:
2837:
2796:
2795:
2789:
2778:
2764:
2725:
2719:
2684:
2683:
2654:
2602:
2583:
2552:
2529:
2509:
2484:
2468:
2440:
2419:
2392:
2364:
2358:
2343:
2323:
2317:
2311:
2288:
2284:
2283:
2275:
2265:
2256:
2235:
2209:
2180:
2160:
2122:
2116:
2104:
2084:
2051:
2039:
2018:
1997:
1985:{\displaystyle n\not \equiv 0{\pmod {p}}}
1964:
1953:
1939:{\displaystyle \omega =e^{2ik\pi /(p-1)}}
1914:
1901:
1889:
1863:
1835:
1834:
1826:
1822:
1821:
1810:
1774:
1763:
1728:
1676:
1656:
1621:
1620:
1600:
1596:
1595:
1590:
1570:
1550:
1530:
1510:
1482:
1481:
1461:
1457:
1456:
1451:
1423:
1422:
1414:
1410:
1409:
1398:
1378:
1359:
1358:
1350:
1346:
1345:
1343:
1052:
1002:
1001:
990:
824:{\displaystyle \chi _{2}(2)=-\omega ^{2}}
815:
790:
784:
754:
741:
731:
715:
699:
677:
671:
650:
644:
610:
587:
564:
538:
515:
376:
370:
309:
263:
257:
199:
5417:
4929:is positive integer. (the definition of
2143:{\displaystyle n\equiv g^{a}{\pmod {p}}}
9122:Knowledge vital articles in Mathematics
2929:which leads to the functional equation
60:
19:
8975:2A00:23C4:7C87:4F00:15D4:7730:972E:49E
1565:can be uniquely written as a power of
9137:C-Class vital articles in Mathematics
624:{\displaystyle \omega =\exp(\pi i/3)}
552:{\displaystyle \omega =\exp(\pi i/6)}
7:
5260:) = 1 are (we write these values in
5063:are coprime positive integers, then
4980:are coprime positive integers, then
2414:is exhibited. And finally by noting
2027:{\displaystyle \chi (n)=\omega ^{a}}
1795:{\displaystyle n\equiv 0{\pmod {p}}}
1446:can be mapped to the additive group
1093:{\displaystyle \chi (k+am)=\chi (k)}
106:This article is within the scope of
8938:{\displaystyle e^{{2\pi i}/(31-1)}}
8570:{\displaystyle e^{{2\pi i}/(29-1)}}
7931:{\displaystyle e^{{2\pi i}/(23-1)}}
7552:{\displaystyle e^{{2\pi i}/(19-1)}}
7361:{\displaystyle e^{{2\pi i}/(17-1)}}
6912:{\displaystyle e^{{2\pi i}/(13-1)}}
6659:{\displaystyle e^{{2\pi i}/(11-1)}}
4968:is divisible by 32 is more complex)
4498:≡ (the smallest primitive root mod
1714:{\displaystyle \chi (n+p)=\chi (n)}
1585:, the exponent being an element of
1393:is prime. the multiplicative group
49:It is of interest to the following
9147:High-priority mathematics articles
6196:{\displaystyle e^{{2\pi i}/(7-1)}}
5975:{\displaystyle e^{{2\pi i}/(5-1)}}
5721:{\displaystyle e^{{2\pi i}/(3-1)}}
5617:{\displaystyle e^{{2\pi i}/(2-1)}}
4494:is the smallest integer such that
3602:{\displaystyle e^{{2\pi i}/(p-1)}}
3161:
3099:
3016:
2973:
2790:
2531:
2504:the discrete fourier transform of
14:
5486:the trivial character (we define
5160:is the smallest positive integer
2131:
1973:
1783:
126:Knowledge:WikiProject Mathematics
9117:Knowledge level-5 vital articles
5415:of these numbers (in degrees)):
129:Template:WikiProject Mathematics
93:
83:
62:
29:
20:
5395:is always 1, we only write the
4130:are positive integers, and gcd(
146:This article has been rated as
9127:C-Class level-5 vital articles
8930:
8918:
8562:
8550:
8371:
8349:
8337:
8334:
8147:
8125:
8113:
8110:
7923:
7911:
7544:
7532:
7353:
7341:
6904:
6892:
6651:
6639:
6410:
6388:
6376:
6373:
6188:
6176:
5967:
5955:
5857:
5835:
5823:
5820:
5713:
5701:
5609:
5597:
5362:
5335:
5279:
5267:
5206:= 1 for all positive integers
5147:
5141:
5114:
5108:
5016:
5010:
4467:
4445:
4433:
4430:
3594:
3582:
3330:
3324:
3318:
3300:
3231:
3222:
3180:
3174:
3122:
3116:
3110:
3058:
3046:
3040:
3034:
2987:
2981:
2951:
2939:
2881:
2872:
2830:
2824:
2813:
2807:
2801:
2758:
2746:
2740:
2734:
2701:
2695:
2689:
2680:
2674:
2665:
2659:
2636:
2630:
2624:
2618:
2576:
2567:
2461:
2455:
2430:
2424:
2385:
2379:
2335:
2329:
2272:
2258:
2191:
2185:
2136:
2125:
2079:the discrete logarithm modulo
2072:{\displaystyle a=\log _{g}(n)}
2066:
2060:
2008:
2002:
1978:
1967:
1931:
1919:
1845:
1818:
1788:
1777:
1739:
1733:
1708:
1702:
1693:
1681:
1631:
1617:
1605:
1592:
1492:
1478:
1466:
1453:
1433:
1406:
1087:
1081:
1072:
1057:
1021:
1009:
882:) may have a period less than
802:
796:
738:
724:
712:
705:
689:
683:
618:
601:
546:
529:
500:09:41, 29 September 2006 (UTC)
489:09:34, 29 September 2006 (UTC)
388:
382:
320:
314:
275:
269:
210:
204:
1:
5318:{\displaystyle re^{i\theta }}
2546:
2306:. Then, the Euler product of
2123:
1965:
1775:
1264:21:17, 3 September 2012 (UTC)
1250:12:10, 3 September 2012 (UTC)
870:restricted by the condition (
443:11:18, 10 February 2006 (UTC)
429:00:43, 10 February 2006 (UTC)
120:and see a list of open tasks.
9142:C-Class mathematics articles
3426:is not coprime, for coprime
958:The principal character mod
419:15:52, 9 February 2006 (UTC)
394:{\displaystyle \chi _{1}(n)}
281:{\displaystyle \chi _{1}(n)}
9098:15:36, 18 August 2021 (UTC)
5153:{\displaystyle \lambda (n)}
5120:{\displaystyle \lambda (n)}
3718:is positive integer. (i.e.
3371:20:12, 2 January 2016 (UTC)
2197:{\displaystyle \varphi (p)}
1224:16:49, 31 August 2012 (UTC)
1198:11:14, 31 August 2012 (UTC)
1151:16:37, 31 August 2012 (UTC)
1128:10:25, 31 August 2012 (UTC)
952:06:02, 31 August 2012 (UTC)
930:01:41, 31 August 2012 (UTC)
904:20:09, 30 August 2012 (UTC)
886:. If so, we say that χ is
847:11:43, 21 August 2020 (UTC)
9163:
9068:14:06, 1 August 2021 (UTC)
5053:Carmichael lambda function
1751:{\displaystyle \chi (n)=0}
853:Definitions of primitivity
332:{\displaystyle \phi (2)=1}
222:{\displaystyle \phi (1)=1}
9044:21:11, 30 June 2021 (UTC)
9020:21:43, 13 June 2021 (UTC)
8998:Draft:Dirichlet character
659:{\displaystyle \chi _{2}}
635:08:04, 19 July 2007 (UTC)
482:character as {1,0,-1,0}.
461:21:21, 26 June 2006 (UTC)
178:11:40, 13 July 2006 (UTC)
145:
78:
57:
8983:11:54, 8 July 2020 (UTC)
5044:{\displaystyle \lambda }
4510:is a primitive root mod
4017:is nonnegative integer,
1320:18:19, 16 May 2013 (UTC)
1297:17:59, 16 May 2013 (UTC)
471:04:04, 9 July 2006 (UTC)
183:Missing character tables
152:project's priority scale
9056:And now it's reverted.
8952:special definition for
7226:special definition for
6232:special definition for
5745:special definition for
5408:{\displaystyle \theta }
1545:then every element of
582:radians this should be
109:WikiProject Mathematics
9112:C-Class vital articles
8939:
8884:
8842:
8771:
8718:
8665:
8612:
8571:
8516:
8474:
8421:
8380:
8300:
8250:
8197:
8156:
8076:
8026:
7973:
7932:
7877:
7835:
7782:
7741:
7688:
7647:
7594:
7553:
7498:
7456:
7403:
7362:
7307:
7175:
7122:
7047:
6994:
6913:
6858:
6788:
6735:
6660:
6605:
6539:
6486:
6419:
6339:
6268:
6197:
6142:
6086:
6033:
5976:
5921:
5866:
5786:
5722:
5667:
5618:
5563:
5515:
5409:
5389:
5369:
5319:
5286:
5246:
5200:
5154:
5121:
5092:
5045:
5025:
4958:
4915:
4858:
4808:
4773:
4734:
4699:
4664:
4629:
4594:
4559:
4476:
4396:
4350:
4315:
4280:
4245:
4206:
4171:
4112:
4059:
4003:
3961:
3903:
3850:
3782:
3747:
3700:
3665:
3603:
3548:
3505:
3466:
3412:
3337:
3103:
2977:
2923:
2794:
2708:
2643:
2518:
2498:
2408:
2300:
2244:
2224:
2198:
2169:
2144:
2093:
2073:
2028:
1986:
1940:
1878:
1852:
1805:choose a generator of
1796:
1752:
1715:
1665:
1638:
1579:
1559:
1539:
1519:
1499:
1440:
1387:
1367:
1094:
1037:
825:
773:
660:
625:
576:
553:
395:
333:
282:
223:
8940:
8885:
8843:
8772:
8719:
8666:
8613:
8572:
8517:
8475:
8422:
8381:
8301:
8251:
8198:
8157:
8077:
8027:
7974:
7933:
7878:
7836:
7783:
7742:
7689:
7648:
7595:
7554:
7499:
7457:
7404:
7363:
7308:
7176:
7123:
7048:
6995:
6914:
6859:
6789:
6736:
6661:
6606:
6540:
6487:
6420:
6340:
6269:
6236:= 8 (the same as the
6198:
6143:
6087:
6034:
5977:
5922:
5867:
5787:
5723:
5668:
5619:
5564:
5516:
5410:
5390:
5370:
5320:
5287:
5247:
5201:
5155:
5122:
5093:
5046:
5026:
4959:
4916:
4859:
4809:
4774:
4735:
4700:
4665:
4630:
4595:
4560:
4490:is positive integer,
4477:
4397:
4351:
4316:
4281:
4246:
4207:
4172:
4113:
4060:
4004:
3962:
3904:
3851:
3783:
3748:
3701:
3666:
3604:
3549:
3506:
3467:
3413:
3338:
3083:
2957:
2924:
2774:
2709:
2644:
2519:
2517:{\displaystyle \chi }
2499:
2409:
2301:
2245:
2225:
2199:
2170:
2145:
2094:
2074:
2029:
1987:
1941:
1879:
1853:
1797:
1753:
1716:
1666:
1639:
1580:
1560:
1540:
1520:
1500:
1441:
1388:
1368:
1095:
1038:
826:
779:, but the table says
774:
661:
626:
577:
575:{\displaystyle 2\pi }
554:
451:Sufficient conditions
396:
334:
283:
224:
36:level-5 vital article
8894:
8859:
8781:
8728:
8675:
8622:
8587:
8526:
8491:
8431:
8396:
8310:
8275:
8207:
8172:
8086:
8051:
7983:
7948:
7887:
7852:
7792:
7757:
7698:
7663:
7604:
7569:
7508:
7473:
7413:
7378:
7317:
7282:
7132:
7097:
7004:
6969:
6868:
6833:
6745:
6710:
6615:
6580:
6496:
6461:
6349:
6314:
6243:
6152:
6117:
6043:
6008:
5931:
5896:
5796:
5761:
5677:
5642:
5573:
5538:
5490:
5399:
5379:
5329:
5296:
5264:
5221:
5168:
5135:
5102:
5067:
5035:
4984:
4933:
4868:
4826:
4783:
4748:
4709:
4674:
4639:
4604:
4569:
4534:
4406:
4364:
4325:
4290:
4255:
4220:
4181:
4146:
4069:
4029:
4021:is positive integer.
3971:
3929:
3921:is positive integer.
3860:
3820:
3812:is positive integer)
3757:
3722:
3675:
3629:
3558:
3523:
3515:is positive integer.
3480:
3441:
3387:
2933:
2718:
2653:
2528:
2508:
2418:
2310:
2255:
2234:
2208:
2179:
2159:
2103:
2083:
2038:
1996:
1952:
1888:
1862:
1809:
1762:
1727:
1675:
1655:
1589:
1569:
1549:
1529:
1509:
1450:
1397:
1377:
1342:
1210:and I changed it to
1104:period smaller than
1051:
989:
783:
670:
643:
586:
563:
514:
369:
339:characters modulo 2:
308:
256:
229:characters modulo 1:
198:
132:mathematics articles
3165:
3020:
2223:{\displaystyle p-1}
1877:{\displaystyle p-1}
1338:Consider the field
878:)=1 the function χ(
559:but as a circle is
8935:
8880:
8838:
8767:
8714:
8661:
8608:
8567:
8512:
8470:
8417:
8376:
8296:
8246:
8193:
8152:
8072:
8022:
7969:
7928:
7873:
7831:
7778:
7737:
7684:
7643:
7590:
7549:
7494:
7452:
7399:
7358:
7303:
7171:
7118:
7043:
6990:
6909:
6854:
6784:
6731:
6656:
6601:
6535:
6482:
6415:
6335:
6264:
6193:
6138:
6082:
6029:
5972:
5917:
5862:
5782:
5718:
5663:
5614:
5559:
5511:
5405:
5385:
5365:
5315:
5282:
5242:
5196:
5150:
5117:
5088:
5041:
5021:
4954:
4911:
4854:
4804:
4769:
4730:
4695:
4660:
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4590:
4555:
4472:
4392:
4346:
4311:
4276:
4241:
4202:
4167:
4108:
4055:
3999:
3957:
3899:
3846:
3778:
3743:
3696:
3661:
3599:
3544:
3501:
3462:
3408:
3349:simple arithmetic,
3333:
3151:
3030:
3006:
2919:
2730:
2704:
2639:
2563:
2547:
2514:
2494:
2451:
2404:
2363:
2322:
2296:
2240:
2220:
2194:
2165:
2140:
2132:
2124:
2089:
2069:
2024:
1982:
1974:
1966:
1936:
1874:
1848:
1792:
1784:
1776:
1748:
1711:
1661:
1634:
1575:
1555:
1535:
1525:is a generator of
1515:
1495:
1436:
1383:
1363:
1331:Simple explanation
1112:always has period
1090:
1033:
821:
769:
656:
621:
572:
549:
391:
329:
278:
219:
101:Mathematics portal
45:content assessment
8985:
8973:comment added by
8960:
8959:
8874:
8832:
8814:
8796:
8761:
8743:
8708:
8690:
8655:
8637:
8602:
8506:
8464:
8446:
8411:
8290:
8240:
8222:
8187:
8066:
8016:
7998:
7963:
7867:
7825:
7807:
7772:
7731:
7713:
7678:
7637:
7619:
7584:
7488:
7446:
7428:
7393:
7297:
7165:
7147:
7112:
7037:
7019:
6984:
6848:
6778:
6760:
6725:
6595:
6529:
6511:
6476:
6329:
6258:
6132:
6076:
6058:
6023:
5911:
5776:
5657:
5553:
5505:
5388:{\displaystyle r}
5375:, and since this
5236:
5184:
5082:
5000:
4948:
4905:
4848:
4798:
4763:
4724:
4689:
4654:
4619:
4584:
4549:
4386:
4340:
4305:
4270:
4235:
4196:
4161:
4102:
4084:
4049:
3987:
3951:
3893:
3875:
3840:
3772:
3737:
3690:
3655:
3538:
3495:
3456:
3402:
3149:
3113:
3021:
2804:
2772:
2721:
2692:
2548:
2436:
2402:
2354:
2349:
2313:
2243:{\displaystyle p}
2168:{\displaystyle p}
2092:{\displaystyle p}
1664:{\displaystyle p}
1578:{\displaystyle g}
1558:{\displaystyle G}
1538:{\displaystyle G}
1518:{\displaystyle g}
1386:{\displaystyle p}
1300:
1283:comment added by
1254:Looks OK to me.
1206:The article said
849:
837:comment added by
410:
409:
301:=== Modulus 2 ===
294:
293:
191:=== Modulus 1 ===
166:
165:
162:
161:
158:
157:
9154:
8989:Proposed Rewrite
8944:
8942:
8941:
8936:
8934:
8933:
8917:
8912:
8889:
8887:
8886:
8881:
8879:
8875:
8867:
8847:
8845:
8844:
8839:
8837:
8833:
8825:
8819:
8815:
8807:
8801:
8797:
8789:
8776:
8774:
8773:
8768:
8766:
8762:
8754:
8748:
8744:
8736:
8723:
8721:
8720:
8715:
8713:
8709:
8701:
8695:
8691:
8683:
8670:
8668:
8667:
8662:
8660:
8656:
8648:
8642:
8638:
8630:
8617:
8615:
8614:
8609:
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8603:
8595:
8576:
8574:
8573:
8568:
8566:
8565:
8549:
8544:
8521:
8519:
8518:
8513:
8511:
8507:
8499:
8479:
8477:
8476:
8471:
8469:
8465:
8457:
8451:
8447:
8439:
8426:
8424:
8423:
8418:
8416:
8412:
8404:
8385:
8383:
8382:
8377:
8375:
8374:
8370:
8369:
8333:
8328:
8305:
8303:
8302:
8297:
8295:
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8283:
8255:
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8245:
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8223:
8215:
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8200:
8199:
8194:
8192:
8188:
8180:
8161:
8159:
8158:
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8145:
8109:
8104:
8081:
8079:
8078:
8073:
8071:
8067:
8059:
8031:
8029:
8028:
8023:
8021:
8017:
8009:
8003:
7999:
7991:
7978:
7976:
7975:
7970:
7968:
7964:
7956:
7937:
7935:
7934:
7929:
7927:
7926:
7910:
7905:
7882:
7880:
7879:
7874:
7872:
7868:
7860:
7840:
7838:
7837:
7832:
7830:
7826:
7818:
7812:
7808:
7800:
7787:
7785:
7784:
7779:
7777:
7773:
7765:
7746:
7744:
7743:
7738:
7736:
7732:
7724:
7718:
7714:
7706:
7693:
7691:
7690:
7685:
7683:
7679:
7671:
7652:
7650:
7649:
7644:
7642:
7638:
7630:
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7620:
7612:
7599:
7597:
7596:
7591:
7589:
7585:
7577:
7558:
7556:
7555:
7550:
7548:
7547:
7531:
7526:
7503:
7501:
7500:
7495:
7493:
7489:
7481:
7461:
7459:
7458:
7453:
7451:
7447:
7439:
7433:
7429:
7421:
7408:
7406:
7405:
7400:
7398:
7394:
7386:
7367:
7365:
7364:
7359:
7357:
7356:
7340:
7335:
7312:
7310:
7309:
7304:
7302:
7298:
7290:
7180:
7178:
7177:
7172:
7170:
7166:
7158:
7152:
7148:
7140:
7127:
7125:
7124:
7119:
7117:
7113:
7105:
7052:
7050:
7049:
7044:
7042:
7038:
7030:
7024:
7020:
7012:
6999:
6997:
6996:
6991:
6989:
6985:
6977:
6918:
6916:
6915:
6910:
6908:
6907:
6891:
6886:
6863:
6861:
6860:
6855:
6853:
6849:
6841:
6793:
6791:
6790:
6785:
6783:
6779:
6771:
6765:
6761:
6753:
6740:
6738:
6737:
6732:
6730:
6726:
6718:
6665:
6663:
6662:
6657:
6655:
6654:
6638:
6633:
6610:
6608:
6607:
6602:
6600:
6596:
6588:
6544:
6542:
6541:
6536:
6534:
6530:
6522:
6516:
6512:
6504:
6491:
6489:
6488:
6483:
6481:
6477:
6469:
6424:
6422:
6421:
6416:
6414:
6413:
6409:
6408:
6372:
6367:
6344:
6342:
6341:
6336:
6334:
6330:
6322:
6273:
6271:
6270:
6265:
6263:
6259:
6251:
6238:kronecker symbol
6202:
6200:
6199:
6194:
6192:
6191:
6175:
6170:
6147:
6145:
6144:
6139:
6137:
6133:
6125:
6091:
6089:
6088:
6083:
6081:
6077:
6069:
6063:
6059:
6051:
6038:
6036:
6035:
6030:
6028:
6024:
6016:
5981:
5979:
5978:
5973:
5971:
5970:
5954:
5949:
5926:
5924:
5923:
5918:
5916:
5912:
5904:
5871:
5869:
5868:
5863:
5861:
5860:
5856:
5855:
5819:
5814:
5791:
5789:
5788:
5783:
5781:
5777:
5769:
5727:
5725:
5724:
5719:
5717:
5716:
5700:
5695:
5672:
5670:
5669:
5664:
5662:
5658:
5650:
5623:
5621:
5620:
5615:
5613:
5612:
5596:
5591:
5568:
5566:
5565:
5560:
5558:
5554:
5546:
5520:
5518:
5517:
5512:
5510:
5506:
5498:
5418:
5414:
5412:
5411:
5406:
5394:
5392:
5391:
5386:
5374:
5372:
5371:
5366:
5324:
5322:
5321:
5316:
5314:
5313:
5291:
5289:
5288:
5285:{\displaystyle }
5283:
5251:
5249:
5248:
5243:
5241:
5237:
5229:
5205:
5203:
5202:
5197:
5195:
5194:
5189:
5185:
5177:
5159:
5157:
5156:
5151:
5126:
5124:
5123:
5118:
5097:
5095:
5094:
5089:
5087:
5083:
5075:
5050:
5048:
5047:
5042:
5030:
5028:
5027:
5022:
5020:
5019:
5005:
5001:
4993:
4963:
4961:
4960:
4955:
4953:
4949:
4941:
4925:≥ 3 is integer,
4920:
4918:
4917:
4912:
4910:
4906:
4904:
4903:
4894:
4887:
4886:
4876:
4863:
4861:
4860:
4855:
4853:
4849:
4847:
4846:
4834:
4813:
4811:
4810:
4805:
4803:
4799:
4791:
4778:
4776:
4775:
4770:
4768:
4764:
4756:
4739:
4737:
4736:
4731:
4729:
4725:
4717:
4704:
4702:
4701:
4696:
4694:
4690:
4682:
4669:
4667:
4666:
4661:
4659:
4655:
4647:
4634:
4632:
4631:
4626:
4624:
4620:
4612:
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4577:
4564:
4562:
4561:
4556:
4554:
4550:
4542:
4481:
4479:
4478:
4473:
4471:
4470:
4466:
4465:
4429:
4424:
4401:
4399:
4398:
4393:
4391:
4387:
4385:
4384:
4372:
4355:
4353:
4352:
4347:
4345:
4341:
4333:
4320:
4318:
4317:
4312:
4310:
4306:
4298:
4285:
4283:
4282:
4277:
4275:
4271:
4263:
4250:
4248:
4247:
4242:
4240:
4236:
4228:
4211:
4209:
4208:
4203:
4201:
4197:
4189:
4176:
4174:
4173:
4168:
4166:
4162:
4154:
4117:
4115:
4114:
4109:
4107:
4103:
4095:
4089:
4085:
4077:
4064:
4062:
4061:
4056:
4054:
4050:
4048:
4037:
4008:
4006:
4005:
4000:
3998:
3997:
3992:
3988:
3980:
3966:
3964:
3963:
3958:
3956:
3952:
3947:
3946:
3937:
3908:
3906:
3905:
3900:
3898:
3894:
3886:
3880:
3876:
3868:
3855:
3853:
3852:
3847:
3845:
3841:
3836:
3828:
3787:
3785:
3784:
3779:
3777:
3773:
3765:
3752:
3750:
3749:
3744:
3742:
3738:
3730:
3705:
3703:
3702:
3697:
3695:
3691:
3683:
3670:
3668:
3667:
3662:
3660:
3656:
3651:
3637:
3608:
3606:
3605:
3600:
3598:
3597:
3581:
3576:
3553:
3551:
3550:
3545:
3543:
3539:
3531:
3510:
3508:
3507:
3502:
3500:
3496:
3488:
3471:
3469:
3468:
3463:
3461:
3457:
3449:
3417:
3415:
3414:
3409:
3407:
3403:
3395:
3342:
3340:
3339:
3334:
3287:
3286:
3274:
3270:
3263:
3262:
3258:
3215:
3214:
3210:
3164:
3159:
3150:
3142:
3140:
3139:
3115:
3114:
3106:
3102:
3097:
3073:
3072:
3029:
3019:
3014:
3002:
3001:
2976:
2971:
2928:
2926:
2925:
2920:
2918:
2914:
2913:
2912:
2908:
2865:
2864:
2860:
2806:
2805:
2797:
2793:
2788:
2773:
2765:
2729:
2713:
2711:
2710:
2705:
2694:
2693:
2685:
2648:
2646:
2645:
2640:
2611:
2610:
2606:
2562:
2523:
2521:
2520:
2515:
2503:
2501:
2500:
2495:
2493:
2492:
2488:
2450:
2413:
2411:
2410:
2405:
2403:
2401:
2400:
2399:
2365:
2362:
2350:
2348:
2347:
2338:
2324:
2321:
2305:
2303:
2302:
2297:
2295:
2294:
2293:
2292:
2287:
2270:
2269:
2249:
2247:
2246:
2241:
2229:
2227:
2226:
2221:
2203:
2201:
2200:
2195:
2174:
2172:
2171:
2166:
2149:
2147:
2146:
2141:
2139:
2121:
2120:
2098:
2096:
2095:
2090:
2078:
2076:
2075:
2070:
2056:
2055:
2033:
2031:
2030:
2025:
2023:
2022:
1991:
1989:
1988:
1983:
1981:
1945:
1943:
1942:
1937:
1935:
1934:
1918:
1883:
1881:
1880:
1875:
1857:
1855:
1854:
1849:
1838:
1830:
1825:
1801:
1799:
1798:
1793:
1791:
1757:
1755:
1754:
1749:
1720:
1718:
1717:
1712:
1670:
1668:
1667:
1662:
1643:
1641:
1640:
1635:
1624:
1604:
1599:
1584:
1582:
1581:
1576:
1564:
1562:
1561:
1556:
1544:
1542:
1541:
1536:
1524:
1522:
1521:
1516:
1504:
1502:
1501:
1496:
1485:
1465:
1460:
1445:
1443:
1442:
1437:
1426:
1418:
1413:
1392:
1390:
1389:
1384:
1372:
1370:
1369:
1364:
1362:
1354:
1349:
1299:
1277:
1099:
1097:
1096:
1091:
1042:
1040:
1039:
1034:
1005:
830:
828:
827:
822:
820:
819:
795:
794:
778:
776:
775:
770:
759:
758:
746:
745:
736:
735:
720:
719:
704:
703:
682:
681:
665:
663:
662:
657:
655:
654:
630:
628:
627:
622:
614:
581:
579:
578:
573:
558:
556:
555:
550:
542:
400:
398:
397:
392:
381:
380:
345:
338:
336:
335:
330:
287:
285:
284:
279:
268:
267:
237:
228:
226:
225:
220:
134:
133:
130:
127:
124:
103:
98:
97:
87:
80:
79:
74:
66:
59:
42:
33:
32:
25:
24:
16:
9162:
9161:
9157:
9156:
9155:
9153:
9152:
9151:
9102:
9101:
9075:
9051:
9027:
8991:
8965:
8897:
8892:
8891:
8862:
8857:
8856:
8820:
8802:
8784:
8779:
8778:
8749:
8731:
8726:
8725:
8696:
8678:
8673:
8672:
8643:
8625:
8620:
8619:
8590:
8585:
8584:
8529:
8524:
8523:
8494:
8489:
8488:
8452:
8434:
8429:
8428:
8399:
8394:
8393:
8355:
8313:
8308:
8307:
8278:
8273:
8272:
8228:
8210:
8205:
8204:
8175:
8170:
8169:
8131:
8089:
8084:
8083:
8054:
8049:
8048:
8004:
7986:
7981:
7980:
7951:
7946:
7945:
7890:
7885:
7884:
7855:
7850:
7849:
7813:
7795:
7790:
7789:
7760:
7755:
7754:
7719:
7701:
7696:
7695:
7666:
7661:
7660:
7625:
7607:
7602:
7601:
7572:
7567:
7566:
7511:
7506:
7505:
7476:
7471:
7470:
7434:
7416:
7411:
7410:
7381:
7376:
7375:
7320:
7315:
7314:
7285:
7280:
7279:
7153:
7135:
7130:
7129:
7100:
7095:
7094:
7025:
7007:
7002:
7001:
6972:
6967:
6966:
6871:
6866:
6865:
6836:
6831:
6830:
6766:
6748:
6743:
6742:
6713:
6708:
6707:
6618:
6613:
6612:
6583:
6578:
6577:
6517:
6499:
6494:
6493:
6464:
6459:
6458:
6394:
6352:
6347:
6346:
6317:
6312:
6311:
6246:
6241:
6240:
6155:
6150:
6149:
6120:
6115:
6114:
6064:
6046:
6041:
6040:
6011:
6006:
6005:
5934:
5929:
5928:
5899:
5894:
5893:
5841:
5799:
5794:
5793:
5764:
5759:
5758:
5680:
5675:
5674:
5645:
5640:
5639:
5576:
5571:
5570:
5541:
5536:
5535:
5493:
5488:
5487:
5397:
5396:
5377:
5376:
5327:
5326:
5302:
5294:
5293:
5262:
5261:
5224:
5219:
5218:
5172:
5171:
5166:
5165:
5133:
5132:
5100:
5099:
5070:
5065:
5064:
5033:
5032:
4988:
4987:
4982:
4981:
4936:
4931:
4930:
4895:
4878:
4877:
4871:
4866:
4865:
4838:
4829:
4824:
4823:
4786:
4781:
4780:
4751:
4746:
4745:
4712:
4707:
4706:
4677:
4672:
4671:
4642:
4637:
4636:
4607:
4602:
4601:
4572:
4567:
4566:
4537:
4532:
4531:
4451:
4409:
4404:
4403:
4376:
4367:
4362:
4361:
4328:
4323:
4322:
4293:
4288:
4287:
4258:
4253:
4252:
4223:
4218:
4217:
4184:
4179:
4178:
4149:
4144:
4143:
4090:
4072:
4067:
4066:
4041:
4032:
4027:
4026:
3975:
3974:
3969:
3968:
3938:
3932:
3927:
3926:
3881:
3863:
3858:
3857:
3829:
3823:
3818:
3817:
3760:
3755:
3754:
3725:
3720:
3719:
3678:
3673:
3672:
3638:
3632:
3627:
3626:
3561:
3556:
3555:
3526:
3521:
3520:
3483:
3478:
3477:
3444:
3439:
3438:
3390:
3385:
3384:
3378:
3355:linear algebra,
3275:
3234:
3183:
3170:
3166:
3125:
3061:
2990:
2931:
2930:
2884:
2833:
2820:
2816:
2716:
2715:
2651:
2650:
2579:
2526:
2525:
2506:
2505:
2464:
2416:
2415:
2388:
2369:
2339:
2325:
2308:
2307:
2282:
2271:
2261:
2253:
2252:
2232:
2231:
2206:
2205:
2177:
2176:
2157:
2156:
2112:
2101:
2100:
2081:
2080:
2047:
2036:
2035:
2014:
1994:
1993:
1950:
1949:
1897:
1886:
1885:
1860:
1859:
1807:
1806:
1760:
1759:
1725:
1724:
1673:
1672:
1653:
1652:
1651:choose a prime
1587:
1586:
1567:
1566:
1547:
1546:
1527:
1526:
1507:
1506:
1448:
1447:
1395:
1394:
1375:
1374:
1340:
1339:
1333:
1278:
1233:is missing iff
1049:
1048:
987:
986:
855:
811:
786:
781:
780:
750:
737:
727:
711:
695:
673:
668:
667:
646:
641:
640:
584:
583:
561:
560:
512:
511:
508:
506:mod 7 character
486:Bird of paradox
479:
477:mod 4 character
453:
372:
367:
366:
306:
305:
259:
254:
253:
196:
195:
185:
171:
131:
128:
125:
122:
121:
99:
92:
72:
43:on Knowledge's
40:
30:
12:
11:
5:
9160:
9158:
9150:
9149:
9144:
9139:
9134:
9129:
9124:
9119:
9114:
9104:
9103:
9074:
9073:Undo Vandalism
9071:
9050:
9047:
9026:
9023:
9010:
9006:
8990:
8987:
8964:
8961:
8958:
8957:
8950:
8946:
8945:
8932:
8929:
8926:
8923:
8920:
8916:
8911:
8908:
8905:
8900:
8878:
8873:
8870:
8865:
8853:
8849:
8848:
8836:
8831:
8828:
8823:
8818:
8813:
8810:
8805:
8800:
8795:
8792:
8787:
8765:
8760:
8757:
8752:
8747:
8742:
8739:
8734:
8712:
8707:
8704:
8699:
8694:
8689:
8686:
8681:
8659:
8654:
8651:
8646:
8641:
8636:
8633:
8628:
8606:
8601:
8598:
8593:
8582:
8578:
8577:
8564:
8561:
8558:
8555:
8552:
8548:
8543:
8540:
8537:
8532:
8510:
8505:
8502:
8497:
8485:
8481:
8480:
8468:
8463:
8460:
8455:
8450:
8445:
8442:
8437:
8415:
8410:
8407:
8402:
8391:
8387:
8386:
8373:
8368:
8365:
8362:
8358:
8354:
8351:
8348:
8345:
8342:
8339:
8336:
8332:
8327:
8324:
8321:
8316:
8294:
8289:
8286:
8281:
8261:
8257:
8256:
8244:
8239:
8236:
8231:
8226:
8221:
8218:
8213:
8191:
8186:
8183:
8178:
8167:
8163:
8162:
8149:
8144:
8141:
8138:
8134:
8130:
8127:
8124:
8121:
8118:
8115:
8112:
8108:
8103:
8100:
8097:
8092:
8070:
8065:
8062:
8057:
8037:
8033:
8032:
8020:
8015:
8012:
8007:
8002:
7997:
7994:
7989:
7967:
7962:
7959:
7954:
7943:
7939:
7938:
7925:
7922:
7919:
7916:
7913:
7909:
7904:
7901:
7898:
7893:
7871:
7866:
7863:
7858:
7846:
7842:
7841:
7829:
7824:
7821:
7816:
7811:
7806:
7803:
7798:
7776:
7771:
7768:
7763:
7752:
7748:
7747:
7735:
7730:
7727:
7722:
7717:
7712:
7709:
7704:
7682:
7677:
7674:
7669:
7658:
7654:
7653:
7641:
7636:
7633:
7628:
7623:
7618:
7615:
7610:
7588:
7583:
7580:
7575:
7564:
7560:
7559:
7546:
7543:
7540:
7537:
7534:
7530:
7525:
7522:
7519:
7514:
7492:
7487:
7484:
7479:
7467:
7463:
7462:
7450:
7445:
7442:
7437:
7432:
7427:
7424:
7419:
7397:
7392:
7389:
7384:
7373:
7369:
7368:
7355:
7352:
7349:
7346:
7343:
7339:
7334:
7331:
7328:
7323:
7301:
7296:
7293:
7288:
7276:
7272:
7271:
7268:
7266:
7263:
7261:
7258:
7256:
7253:
7251:
7248:
7246:
7243:
7241:
7238:
7236:
7233:
7231:
7224:
7220:
7219:
7216:
7213:
7211:
7208:
7206:
7204:
7201:
7198:
7196:
7194:
7191:
7189:
7186:
7183:
7181:
7169:
7164:
7161:
7156:
7151:
7146:
7143:
7138:
7116:
7111:
7108:
7103:
7092:
7088:
7087:
7084:
7082:
7079:
7077:
7074:
7072:
7070:
7068:
7065:
7063:
7060:
7058:
7055:
7053:
7041:
7036:
7033:
7028:
7023:
7018:
7015:
7010:
6988:
6983:
6980:
6975:
6964:
6960:
6959:
6956:
6953:
6950:
6947:
6944:
6941:
6938:
6935:
6932:
6929:
6924:
6921:
6919:
6906:
6903:
6900:
6897:
6894:
6890:
6885:
6882:
6879:
6874:
6852:
6847:
6844:
6839:
6827:
6823:
6822:
6819:
6817:
6815:
6813:
6810:
6808:
6805:
6803:
6801:
6799:
6796:
6794:
6782:
6777:
6774:
6769:
6764:
6759:
6756:
6751:
6729:
6724:
6721:
6716:
6705:
6701:
6700:
6697:
6694:
6691:
6688:
6685:
6682:
6679:
6676:
6671:
6668:
6666:
6653:
6650:
6647:
6644:
6641:
6637:
6632:
6629:
6626:
6621:
6599:
6594:
6591:
6586:
6574:
6570:
6569:
6566:
6564:
6561:
6559:
6557:
6555:
6552:
6550:
6547:
6545:
6533:
6528:
6525:
6520:
6515:
6510:
6507:
6502:
6480:
6475:
6472:
6467:
6456:
6452:
6451:
6448:
6445:
6443:
6440:
6437:
6435:
6430:
6427:
6425:
6412:
6407:
6404:
6401:
6397:
6393:
6390:
6387:
6384:
6381:
6378:
6375:
6371:
6366:
6363:
6360:
6355:
6333:
6328:
6325:
6320:
6300:
6296:
6295:
6292:
6290:
6287:
6285:
6282:
6280:
6277:
6275:
6262:
6257:
6254:
6249:
6230:
6226:
6225:
6222:
6219:
6216:
6211:
6208:
6205:
6203:
6190:
6187:
6184:
6181:
6178:
6174:
6169:
6166:
6163:
6158:
6136:
6131:
6128:
6123:
6111:
6107:
6106:
6103:
6101:
6099:
6097:
6094:
6092:
6080:
6075:
6072:
6067:
6062:
6057:
6054:
6049:
6027:
6022:
6019:
6014:
6003:
5999:
5998:
5995:
5992:
5987:
5984:
5982:
5969:
5966:
5963:
5960:
5957:
5953:
5948:
5945:
5942:
5937:
5915:
5910:
5907:
5902:
5890:
5886:
5885:
5880:
5878:
5875:
5873:
5859:
5854:
5851:
5848:
5844:
5840:
5837:
5834:
5831:
5828:
5825:
5822:
5818:
5813:
5810:
5807:
5802:
5780:
5775:
5772:
5767:
5743:
5739:
5738:
5733:
5730:
5728:
5715:
5712:
5709:
5706:
5703:
5699:
5694:
5691:
5688:
5683:
5661:
5656:
5653:
5648:
5636:
5632:
5631:
5626:
5624:
5611:
5608:
5605:
5602:
5599:
5595:
5590:
5587:
5584:
5579:
5557:
5552:
5549:
5544:
5532:
5528:
5527:
5522:
5509:
5504:
5501:
5496:
5484:
5480:
5479:
5476:
5473:
5470:
5467:
5464:
5461:
5458:
5455:
5452:
5449:
5446:
5443:
5440:
5437:
5434:
5431:
5428:
5404:
5384:
5364:
5361:
5358:
5355:
5352:
5349:
5346:
5343:
5340:
5337:
5334:
5312:
5309:
5305:
5301:
5281:
5278:
5275:
5272:
5269:
5240:
5235:
5232:
5227:
5217:The values of
5193:
5188:
5183:
5180:
5175:
5149:
5146:
5143:
5140:
5116:
5113:
5110:
5107:
5086:
5081:
5078:
5073:
5040:
5018:
5015:
5012:
5009:
5004:
4999:
4996:
4991:
4970:
4969:
4952:
4947:
4944:
4939:
4909:
4902:
4898:
4893:
4890:
4885:
4881:
4874:
4852:
4845:
4841:
4837:
4832:
4820:
4819:
4802:
4797:
4794:
4789:
4767:
4762:
4759:
4754:
4728:
4723:
4720:
4715:
4693:
4688:
4685:
4680:
4658:
4653:
4650:
4645:
4623:
4618:
4615:
4610:
4588:
4583:
4580:
4575:
4553:
4548:
4545:
4540:
4528:
4527:
4486:is odd prime,
4469:
4464:
4461:
4458:
4454:
4450:
4447:
4444:
4441:
4438:
4435:
4432:
4428:
4423:
4420:
4417:
4412:
4390:
4383:
4379:
4375:
4370:
4358:
4357:
4344:
4339:
4336:
4331:
4309:
4304:
4301:
4296:
4274:
4269:
4266:
4261:
4239:
4234:
4231:
4226:
4214:
4213:
4200:
4195:
4192:
4187:
4165:
4160:
4157:
4152:
4140:
4139:
4106:
4101:
4098:
4093:
4088:
4083:
4080:
4075:
4053:
4047:
4044:
4040:
4035:
4023:
4022:
3996:
3991:
3986:
3983:
3978:
3955:
3950:
3945:
3941:
3935:
3923:
3922:
3917:are integers,
3897:
3892:
3889:
3884:
3879:
3874:
3871:
3866:
3844:
3839:
3835:
3832:
3826:
3814:
3813:
3808:are integers,
3776:
3771:
3768:
3763:
3741:
3736:
3733:
3728:
3714:are integers,
3694:
3689:
3686:
3681:
3659:
3654:
3650:
3647:
3644:
3641:
3635:
3623:
3622:
3613:is prime, and
3596:
3593:
3590:
3587:
3584:
3580:
3575:
3572:
3569:
3564:
3542:
3537:
3534:
3529:
3517:
3516:
3499:
3494:
3491:
3486:
3474:
3473:
3460:
3455:
3452:
3447:
3406:
3401:
3398:
3393:
3377:
3374:
3360:
3359:
3356:
3353:
3350:
3332:
3329:
3326:
3323:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3285:
3282:
3278:
3273:
3269:
3266:
3261:
3257:
3253:
3250:
3247:
3244:
3241:
3237:
3233:
3230:
3227:
3224:
3221:
3218:
3213:
3209:
3205:
3202:
3199:
3196:
3193:
3190:
3186:
3182:
3179:
3176:
3173:
3169:
3163:
3158:
3154:
3148:
3145:
3138:
3135:
3132:
3128:
3124:
3121:
3118:
3112:
3109:
3101:
3096:
3093:
3090:
3086:
3082:
3079:
3076:
3071:
3068:
3064:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3028:
3024:
3018:
3013:
3009:
3005:
3000:
2997:
2993:
2989:
2986:
2983:
2980:
2975:
2970:
2967:
2964:
2960:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2917:
2911:
2907:
2903:
2900:
2897:
2894:
2891:
2887:
2883:
2880:
2877:
2874:
2871:
2868:
2863:
2859:
2855:
2852:
2849:
2846:
2843:
2840:
2836:
2832:
2829:
2826:
2823:
2819:
2815:
2812:
2809:
2803:
2800:
2792:
2787:
2784:
2781:
2777:
2771:
2768:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2733:
2728:
2724:
2703:
2700:
2697:
2691:
2688:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2609:
2605:
2601:
2598:
2595:
2592:
2589:
2586:
2582:
2578:
2575:
2572:
2569:
2566:
2561:
2558:
2555:
2551:
2545:
2542:
2539:
2536:
2533:
2513:
2491:
2487:
2483:
2480:
2477:
2474:
2471:
2467:
2463:
2460:
2457:
2454:
2449:
2446:
2443:
2439:
2435:
2432:
2429:
2426:
2423:
2398:
2395:
2391:
2387:
2384:
2381:
2378:
2375:
2372:
2368:
2361:
2357:
2353:
2346:
2342:
2337:
2334:
2331:
2328:
2320:
2316:
2291:
2286:
2281:
2278:
2274:
2268:
2264:
2260:
2239:
2219:
2216:
2213:
2193:
2190:
2187:
2184:
2164:
2152:
2151:
2138:
2135:
2130:
2127:
2119:
2115:
2111:
2108:
2088:
2068:
2065:
2062:
2059:
2054:
2050:
2046:
2043:
2021:
2017:
2013:
2010:
2007:
2004:
2001:
1980:
1977:
1972:
1969:
1963:
1960:
1957:
1946:
1933:
1930:
1927:
1924:
1921:
1917:
1913:
1910:
1907:
1904:
1900:
1896:
1893:
1884:root of unity
1873:
1870:
1867:
1847:
1844:
1841:
1837:
1833:
1829:
1824:
1820:
1817:
1814:
1803:
1790:
1787:
1782:
1779:
1773:
1770:
1767:
1747:
1744:
1741:
1738:
1735:
1732:
1721:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1660:
1633:
1630:
1627:
1623:
1619:
1616:
1613:
1610:
1607:
1603:
1598:
1594:
1574:
1554:
1534:
1514:
1494:
1491:
1488:
1484:
1480:
1477:
1474:
1471:
1468:
1464:
1459:
1455:
1435:
1432:
1429:
1425:
1421:
1417:
1412:
1408:
1405:
1402:
1382:
1361:
1357:
1353:
1348:
1332:
1329:
1328:
1327:
1326:
1325:
1273:
1272:
1271:
1270:
1269:
1268:
1267:
1266:
1201:
1200:
1156:
1155:
1154:
1153:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1004:
1000:
997:
994:
966:such that gcd(
956:
955:
954:
933:
932:
890:and otherwise
854:
851:
818:
814:
810:
807:
804:
801:
798:
793:
789:
768:
765:
762:
757:
753:
749:
744:
740:
734:
730:
726:
723:
718:
714:
710:
707:
702:
698:
694:
691:
688:
685:
680:
676:
653:
649:
620:
617:
613:
609:
606:
603:
600:
597:
594:
591:
571:
568:
548:
545:
541:
537:
534:
531:
528:
525:
522:
519:
507:
504:
503:
502:
478:
475:
474:
473:
452:
449:
448:
447:
446:
445:
432:
431:
412:
411:
408:
407:
404:
401:
390:
387:
384:
379:
375:
363:
362:
357:
352:
341:
340:
328:
325:
322:
319:
316:
313:
302:
299:
296:
295:
292:
291:
288:
277:
274:
271:
266:
262:
250:
249:
244:
233:
232:
230:
218:
215:
212:
209:
206:
203:
192:
184:
181:
170:
167:
164:
163:
160:
159:
156:
155:
144:
138:
137:
135:
118:the discussion
105:
104:
88:
76:
75:
67:
55:
54:
48:
26:
13:
10:
9:
6:
4:
3:
2:
9159:
9148:
9145:
9143:
9140:
9138:
9135:
9133:
9130:
9128:
9125:
9123:
9120:
9118:
9115:
9113:
9110:
9109:
9107:
9100:
9099:
9095:
9091:
9087:
9084:
9081:
9078:
9072:
9070:
9069:
9065:
9061:
9057:
9054:
9048:
9046:
9045:
9041:
9037:
9033:
9030:
9024:
9022:
9021:
9017:
9013:
9008:
9004:
9001:
8999:
8994:
8988:
8986:
8984:
8980:
8976:
8972:
8962:
8955:
8951:
8948:
8947:
8927:
8924:
8921:
8914:
8909:
8906:
8903:
8898:
8876:
8871:
8868:
8863:
8854:
8851:
8850:
8834:
8829:
8826:
8821:
8816:
8811:
8808:
8803:
8798:
8793:
8790:
8785:
8763:
8758:
8755:
8750:
8745:
8740:
8737:
8732:
8710:
8705:
8702:
8697:
8692:
8687:
8684:
8679:
8657:
8652:
8649:
8644:
8639:
8634:
8631:
8626:
8604:
8599:
8596:
8591:
8583:
8580:
8579:
8559:
8556:
8553:
8546:
8541:
8538:
8535:
8530:
8508:
8503:
8500:
8495:
8486:
8483:
8482:
8466:
8461:
8458:
8453:
8448:
8443:
8440:
8435:
8413:
8408:
8405:
8400:
8392:
8389:
8388:
8366:
8363:
8360:
8356:
8352:
8346:
8343:
8340:
8330:
8325:
8322:
8319:
8314:
8292:
8287:
8284:
8279:
8270:
8266:
8262:
8259:
8258:
8242:
8237:
8234:
8229:
8224:
8219:
8216:
8211:
8189:
8184:
8181:
8176:
8168:
8165:
8164:
8142:
8139:
8136:
8132:
8128:
8122:
8119:
8116:
8106:
8101:
8098:
8095:
8090:
8068:
8063:
8060:
8055:
8046:
8042:
8038:
8035:
8034:
8018:
8013:
8010:
8005:
8000:
7995:
7992:
7987:
7965:
7960:
7957:
7952:
7944:
7941:
7940:
7920:
7917:
7914:
7907:
7902:
7899:
7896:
7891:
7869:
7864:
7861:
7856:
7847:
7844:
7843:
7827:
7822:
7819:
7814:
7809:
7804:
7801:
7796:
7774:
7769:
7766:
7761:
7753:
7750:
7749:
7733:
7728:
7725:
7720:
7715:
7710:
7707:
7702:
7680:
7675:
7672:
7667:
7659:
7656:
7655:
7639:
7634:
7631:
7626:
7621:
7616:
7613:
7608:
7586:
7581:
7578:
7573:
7565:
7562:
7561:
7541:
7538:
7535:
7528:
7523:
7520:
7517:
7512:
7490:
7485:
7482:
7477:
7468:
7465:
7464:
7448:
7443:
7440:
7435:
7430:
7425:
7422:
7417:
7395:
7390:
7387:
7382:
7374:
7371:
7370:
7350:
7347:
7344:
7337:
7332:
7329:
7326:
7321:
7299:
7294:
7291:
7286:
7277:
7274:
7273:
7269:
7267:
7264:
7262:
7259:
7257:
7254:
7252:
7249:
7247:
7244:
7242:
7239:
7237:
7234:
7232:
7229:
7225:
7222:
7221:
7217:
7214:
7212:
7209:
7207:
7205:
7202:
7199:
7197:
7195:
7192:
7190:
7187:
7184:
7182:
7167:
7162:
7159:
7154:
7149:
7144:
7141:
7136:
7114:
7109:
7106:
7101:
7093:
7090:
7089:
7085:
7083:
7080:
7078:
7075:
7073:
7071:
7069:
7066:
7064:
7061:
7059:
7056:
7054:
7039:
7034:
7031:
7026:
7021:
7016:
7013:
7008:
6986:
6981:
6978:
6973:
6965:
6962:
6961:
6957:
6954:
6951:
6948:
6945:
6942:
6939:
6936:
6933:
6930:
6928:
6925:
6922:
6920:
6901:
6898:
6895:
6888:
6883:
6880:
6877:
6872:
6850:
6845:
6842:
6837:
6828:
6825:
6824:
6820:
6818:
6816:
6814:
6811:
6809:
6806:
6804:
6802:
6800:
6797:
6795:
6780:
6775:
6772:
6767:
6762:
6757:
6754:
6749:
6727:
6722:
6719:
6714:
6706:
6703:
6702:
6698:
6695:
6692:
6689:
6686:
6683:
6680:
6677:
6675:
6672:
6669:
6667:
6648:
6645:
6642:
6635:
6630:
6627:
6624:
6619:
6597:
6592:
6589:
6584:
6575:
6572:
6571:
6567:
6565:
6562:
6560:
6558:
6556:
6553:
6551:
6548:
6546:
6531:
6526:
6523:
6518:
6513:
6508:
6505:
6500:
6478:
6473:
6470:
6465:
6457:
6454:
6453:
6449:
6446:
6444:
6441:
6438:
6436:
6434:
6431:
6428:
6426:
6405:
6402:
6399:
6395:
6391:
6385:
6382:
6379:
6369:
6364:
6361:
6358:
6353:
6331:
6326:
6323:
6318:
6309:
6305:
6301:
6298:
6297:
6293:
6291:
6288:
6286:
6283:
6281:
6278:
6276:
6260:
6255:
6252:
6247:
6239:
6235:
6231:
6228:
6227:
6223:
6220:
6217:
6215:
6212:
6209:
6206:
6204:
6185:
6182:
6179:
6172:
6167:
6164:
6161:
6156:
6134:
6129:
6126:
6121:
6112:
6109:
6108:
6104:
6102:
6100:
6098:
6095:
6093:
6078:
6073:
6070:
6065:
6060:
6055:
6052:
6047:
6025:
6020:
6017:
6012:
6004:
6001:
6000:
5996:
5993:
5991:
5988:
5985:
5983:
5964:
5961:
5958:
5951:
5946:
5943:
5940:
5935:
5913:
5908:
5905:
5900:
5891:
5888:
5887:
5884:
5881:
5879:
5876:
5874:
5852:
5849:
5846:
5842:
5838:
5832:
5829:
5826:
5816:
5811:
5808:
5805:
5800:
5778:
5773:
5770:
5765:
5756:
5752:
5748:
5744:
5741:
5740:
5737:
5734:
5731:
5729:
5710:
5707:
5704:
5697:
5692:
5689:
5686:
5681:
5659:
5654:
5651:
5646:
5637:
5634:
5633:
5630:
5627:
5625:
5606:
5603:
5600:
5593:
5588:
5585:
5582:
5577:
5555:
5550:
5547:
5542:
5533:
5530:
5529:
5526:
5523:
5507:
5502:
5499:
5494:
5485:
5482:
5481:
5477:
5474:
5471:
5468:
5465:
5462:
5459:
5456:
5453:
5450:
5447:
5444:
5441:
5438:
5435:
5432:
5429:
5427:
5423:
5420:
5419:
5416:
5402:
5382:
5359:
5356:
5353:
5350:
5347:
5344:
5341:
5338:
5332:
5310:
5307:
5303:
5299:
5276:
5273:
5270:
5259:
5255:
5238:
5233:
5230:
5225:
5215:
5213:
5209:
5191:
5186:
5181:
5178:
5173:
5163:
5144:
5138:
5130:
5111:
5105:
5084:
5079:
5076:
5071:
5062:
5058:
5054:
5038:
5013:
5007:
5002:
4997:
4994:
4989:
4979:
4975:
4967:
4950:
4945:
4942:
4937:
4928:
4924:
4907:
4900:
4896:
4891:
4888:
4883:
4879:
4872:
4850:
4843:
4839:
4835:
4830:
4822:
4821:
4817:
4800:
4795:
4792:
4787:
4765:
4760:
4757:
4752:
4743:
4726:
4721:
4718:
4713:
4691:
4686:
4683:
4678:
4656:
4651:
4648:
4643:
4621:
4616:
4613:
4608:
4586:
4581:
4578:
4573:
4551:
4546:
4543:
4538:
4530:
4529:
4525:
4521:
4517:
4513:
4509:
4505:
4501:
4497:
4493:
4489:
4485:
4462:
4459:
4456:
4452:
4448:
4442:
4439:
4436:
4426:
4421:
4418:
4415:
4410:
4388:
4381:
4377:
4373:
4368:
4360:
4359:
4342:
4337:
4334:
4329:
4307:
4302:
4299:
4294:
4272:
4267:
4264:
4259:
4237:
4232:
4229:
4224:
4216:
4215:
4198:
4193:
4190:
4185:
4163:
4158:
4155:
4150:
4142:
4141:
4137:
4133:
4129:
4125:
4121:
4104:
4099:
4096:
4091:
4086:
4081:
4078:
4073:
4051:
4045:
4042:
4038:
4033:
4025:
4024:
4020:
4016:
4012:
3994:
3989:
3984:
3981:
3976:
3953:
3948:
3943:
3939:
3933:
3925:
3924:
3920:
3916:
3912:
3895:
3890:
3887:
3882:
3877:
3872:
3869:
3864:
3842:
3837:
3833:
3830:
3824:
3816:
3815:
3811:
3807:
3803:
3799:
3795:
3791:
3774:
3769:
3766:
3761:
3739:
3734:
3731:
3726:
3717:
3713:
3709:
3692:
3687:
3684:
3679:
3657:
3652:
3648:
3645:
3642:
3639:
3633:
3625:
3624:
3620:
3616:
3612:
3591:
3588:
3585:
3578:
3573:
3570:
3567:
3562:
3540:
3535:
3532:
3527:
3519:
3518:
3514:
3497:
3492:
3489:
3484:
3476:
3475:
3458:
3453:
3450:
3445:
3437:
3436:
3435:
3433:
3429:
3425:
3421:
3404:
3399:
3396:
3391:
3381:
3376:New character
3375:
3373:
3372:
3368:
3364:
3363:78.227.78.135
3357:
3354:
3352:group theory,
3351:
3348:
3347:
3346:
3343:
3327:
3321:
3315:
3312:
3309:
3306:
3303:
3297:
3294:
3291:
3288:
3283:
3280:
3276:
3271:
3267:
3264:
3259:
3255:
3251:
3248:
3245:
3242:
3239:
3235:
3228:
3225:
3219:
3216:
3211:
3207:
3203:
3200:
3197:
3194:
3191:
3188:
3184:
3177:
3171:
3167:
3156:
3152:
3146:
3143:
3136:
3133:
3130:
3126:
3119:
3107:
3094:
3091:
3088:
3084:
3080:
3077:
3074:
3069:
3066:
3062:
3055:
3052:
3049:
3043:
3037:
3031:
3026:
3022:
3011:
3007:
3003:
2998:
2995:
2991:
2984:
2978:
2968:
2965:
2962:
2958:
2954:
2948:
2945:
2942:
2936:
2915:
2909:
2905:
2901:
2898:
2895:
2892:
2889:
2885:
2878:
2875:
2869:
2866:
2861:
2857:
2853:
2850:
2847:
2844:
2841:
2838:
2834:
2827:
2821:
2817:
2810:
2798:
2785:
2782:
2779:
2775:
2769:
2766:
2761:
2755:
2752:
2749:
2743:
2737:
2731:
2726:
2722:
2698:
2686:
2677:
2671:
2668:
2662:
2656:
2633:
2627:
2621:
2615:
2612:
2607:
2603:
2599:
2596:
2593:
2590:
2587:
2584:
2580:
2573:
2570:
2564:
2559:
2556:
2553:
2549:
2543:
2540:
2537:
2534:
2511:
2489:
2485:
2481:
2478:
2475:
2472:
2469:
2465:
2458:
2452:
2447:
2444:
2441:
2437:
2433:
2427:
2421:
2396:
2393:
2389:
2382:
2376:
2373:
2370:
2366:
2359:
2355:
2351:
2344:
2340:
2332:
2326:
2318:
2314:
2289:
2279:
2276:
2266:
2262:
2237:
2217:
2214:
2211:
2188:
2182:
2162:
2133:
2128:
2117:
2113:
2109:
2106:
2086:
2063:
2057:
2052:
2048:
2044:
2041:
2019:
2015:
2011:
2005:
1999:
1975:
1970:
1961:
1958:
1955:
1947:
1928:
1925:
1922:
1915:
1911:
1908:
1905:
1902:
1898:
1894:
1891:
1871:
1868:
1865:
1842:
1839:
1831:
1827:
1815:
1812:
1804:
1785:
1780:
1771:
1768:
1765:
1745:
1742:
1736:
1730:
1722:
1705:
1699:
1696:
1690:
1687:
1684:
1678:
1658:
1650:
1649:
1648:
1645:
1628:
1625:
1614:
1611:
1608:
1601:
1572:
1552:
1532:
1512:
1489:
1486:
1475:
1472:
1469:
1462:
1430:
1427:
1419:
1415:
1403:
1400:
1380:
1355:
1351:
1336:
1330:
1323:
1322:
1321:
1318:
1315:
1311:
1307:
1303:
1302:
1301:
1298:
1294:
1290:
1286:
1282:
1265:
1261:
1257:
1253:
1252:
1251:
1248:
1245:
1241:
1236:
1232:
1227:
1226:
1225:
1221:
1217:
1213:
1209:
1205:
1204:
1203:
1202:
1199:
1196:
1193:
1189:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1157:
1152:
1148:
1144:
1139:
1135:
1131:
1130:
1129:
1126:
1123:
1119:
1115:
1111:
1107:
1103:
1084:
1078:
1075:
1069:
1066:
1063:
1060:
1054:
1046:
1027:
1024:
1018:
1015:
1012:
1006:
998:
995:
984:
981:
977:
973:
969:
965:
961:
957:
953:
949:
945:
941:
937:
936:
935:
934:
931:
927:
923:
919:
916:
912:
908:
907:
906:
905:
901:
897:
893:
889:
885:
881:
877:
873:
869:
864:
860:
852:
850:
848:
844:
840:
839:31.125.12.109
836:
816:
812:
808:
805:
799:
791:
787:
766:
763:
760:
755:
751:
747:
742:
732:
728:
721:
716:
708:
700:
696:
692:
686:
678:
674:
651:
647:
637:
636:
633:
615:
611:
607:
604:
598:
595:
592:
589:
569:
566:
543:
539:
535:
532:
526:
523:
520:
517:
505:
501:
498:
493:
492:
491:
490:
487:
483:
476:
472:
469:
465:
464:
463:
462:
459:
450:
444:
441:
436:
435:
434:
433:
430:
427:
423:
422:
421:
420:
417:
405:
402:
385:
377:
373:
365:
364:
361:
358:
356:
353:
351:
347:
346:
343:
342:
326:
323:
317:
311:
303:
300:
298:
297:
289:
272:
264:
260:
252:
251:
248:
245:
243:
239:
238:
235:
234:
231:
216:
213:
207:
201:
193:
190:
189:
188:
182:
180:
179:
176:
168:
153:
149:
148:High-priority
143:
140:
139:
136:
119:
115:
111:
110:
102:
96:
91:
89:
86:
82:
81:
77:
73:High‑priority
71:
68:
65:
61:
56:
52:
46:
38:
37:
27:
23:
18:
17:
9088:
9085:
9082:
9079:
9076:
9058:
9055:
9052:
9034:
9031:
9028:
9009:
9005:
9002:
8995:
8992:
8969:— Preceding
8966:
8953:
8268:
8264:
8044:
8040:
7227:
6926:
6673:
6432:
6307:
6303:
6233:
6213:
5989:
5882:
5754:
5750:
5746:
5735:
5628:
5524:
5425:
5421:
5257:
5253:
5216:
5211:
5207:
5161:
5131:). Besides,
5060:
5056:
4977:
4973:
4971:
4965:
4926:
4922:
4815:
4741:
4523:
4519:
4515:
4511:
4507:
4503:
4499:
4495:
4491:
4487:
4483:
4135:
4131:
4127:
4123:
4122:is integer,
4119:
4018:
4014:
4013:is integer,
4010:
3918:
3914:
3910:
3809:
3805:
3801:
3797:
3793:
3789:
3715:
3711:
3707:
3618:
3614:
3610:
3512:
3431:
3427:
3423:
3419:
3382:
3379:
3361:
3344:
2524:we get that
2153:
1646:
1337:
1334:
1309:
1305:
1279:— Preceding
1274:
1239:
1234:
1230:
1211:
1207:
1187:
1184:
1180:
1176:
1172:
1168:
1164:
1160:
1137:
1133:
1117:
1113:
1109:
1105:
1101:
1044:
982:
979:
975:
971:
967:
963:
959:
939:
917:
914:
910:
891:
887:
883:
879:
875:
871:
867:
858:
856:
833:— Preceding
638:
509:
484:
480:
454:
413:
359:
354:
349:
246:
241:
186:
172:
147:
107:
51:WikiProjects
34:
9090:James in dc
9060:James in dc
9036:James in dc
9032:"Be bold!"
9012:James in dc
5210:coprime to
5031:= 1, where
3511:= 1, where
3434:we define:
1285:Cjs cjs cjs
1256:Deltahedron
1216:Deltahedron
1143:Deltahedron
944:Deltahedron
896:Deltahedron
888:imprimitive
495:character.
123:Mathematics
114:mathematics
70:Mathematics
9106:Categories
8267:such that
8043:such that
6306:such that
5753:such that
5164:such that
3383:We define
863:User:EmilJ
497:Madmath789
458:Madmath789
304:There are
194:There are
9049:Vandalism
9025:Rewritten
8963:Extension
5129:root of 1
5055:(i.e. if
4972:Thus, if
892:primitive
39:is rated
8971:unsigned
5430:comment
5252:for gcd(
4921:, where
4526:= 40487)
4482:, where
4118:, where
4009:, where
3909:, where
3800:, where
3706:, where
3609:, where
2649:so that
1858:, and a
1293:contribs
1281:unsigned
857:I wrote
835:unsigned
9086:Thanks
9007:Thanks
5051:is the
3418:= 0 if
2099:, i.e.
666:. Then
632:Wilmot1
169:History
150:on the
41:C-class
9029:Done.
4964:where
4670:= −1,
4506:, and
4502:) mod
4138:) = 1.
2204:, i.e
2034:where
1373:where
922:RobHar
47:scale.
8956:= 32
7230:= 16
5521:= 1)
5098:is a
4600:= 1,
4356:= −1.
4286:= 1,
4212:= −1.
4177:= 1,
1505:: if
468:linas
426:linas
175:Ninte
28:This
9094:talk
9064:talk
9040:talk
9016:talk
8979:talk
7260:270
7255:180
7250:180
7245:270
7215:270
7210:180
7193:180
7188:270
7086:180
7081:240
7076:120
7067:300
6958:180
6955:210
6952:300
6949:240
6943:330
6940:150
6937:270
6931:120
6812:180
6807:180
6699:180
6696:216
6693:108
6690:252
6687:324
6684:144
6678:288
6568:180
6554:270
6450:180
6447:240
6442:300
6439:120
6289:180
6284:180
6224:180
6221:300
6218:240
6210:120
6105:180
5997:180
5994:270
5127:-th
5059:and
4976:and
4126:and
3913:and
3804:and
3796:mod
3710:and
3472:= 1.
3430:and
3422:and
3367:talk
1992:set
1948:for
1723:set
1314:Emil
1289:talk
1260:talk
1244:Emil
1220:talk
1192:Emil
1171:) =
1147:talk
1122:Emil
948:talk
926:talk
900:talk
843:talk
348:χ \
240:χ \
142:High
8949:32
8852:31
8581:30
8484:29
8390:28
8260:27
8166:26
8036:25
7942:24
7845:23
7751:22
7657:21
7563:20
7466:19
7372:18
7275:17
7265:90
7240:90
7223:16
7203:90
7200:90
7091:15
7062:60
6963:14
6946:90
6934:60
6826:13
6704:12
6681:72
6573:11
6563:90
6455:10
5883:180
5736:180
5478:15
5475:14
5472:13
5469:12
5466:11
5463:10
5354:sin
5339:cos
4814:= −
4514:. (
3788:if
2230:if
2129:mod
2049:log
1971:mod
1781:mod
1758:if
1644:.
1175:(ψ,
1167:(χ,
1047:if
942:.
596:exp
524:exp
9108::
9096:)
9066:)
9042:)
9018:)
9000:.
8981:)
8925:−
8922:31
8907:π
8890:=
8872:31
8777:=
8724:=
8706:10
8671:=
8653:15
8618:=
8600:30
8557:−
8554:29
8539:π
8522:=
8504:29
8427:=
8409:28
8364:−
8353:⋅
8344:−
8323:π
8306:=
8288:27
8238:13
8203:=
8185:26
8140:−
8129:⋅
8120:−
8099:π
8082:=
8064:25
7979:=
7961:24
7918:−
7915:23
7900:π
7883:=
7865:23
7823:11
7788:=
7770:22
7694:=
7676:21
7600:=
7582:20
7539:−
7536:19
7521:π
7504:=
7486:19
7409:=
7391:18
7348:−
7345:17
7330:π
7313:=
7295:17
7270:0
7235:0
7218:0
7185:0
7128:=
7110:15
7057:0
7000:=
6982:14
6927:30
6923:0
6899:−
6896:13
6881:π
6864:=
6846:13
6821:0
6798:0
6741:=
6723:12
6674:36
6670:0
6646:−
6643:11
6628:π
6611:=
6593:11
6549:0
6492:=
6474:10
6433:60
6429:0
6403:−
6392:⋅
6383:−
6362:π
6345:=
6299:9
6294:0
6279:0
6274:)
6229:8
6214:60
6207:0
6183:−
6165:π
6148:=
6110:7
6096:0
6039:=
6002:6
5990:90
5986:0
5962:−
5944:π
5927:=
5889:5
5877:0
5872:)
5850:−
5839:⋅
5830:−
5809:π
5792:=
5742:4
5732:0
5708:−
5690:π
5673:=
5635:3
5604:−
5586:π
5569:=
5531:2
5483:1
5460:9
5457:8
5454:7
5451:6
5448:5
5445:4
5442:3
5439:2
5436:1
5433:0
5424:\
5403:θ
5360:θ
5357:
5345:θ
5342:
5325:=
5311:θ
5292:=
5277:θ
5256:,
5214:.
5139:λ
5106:λ
5039:λ
5008:λ
4889:−
4864:=
4796:16
4793:11
4779:=
4761:16
4744:,
4740:=
4722:16
4719:13
4705:=
4687:16
4652:16
4635:=
4617:16
4582:16
4579:15
4565:=
4547:16
4460:−
4449:⋅
4440:−
4419:π
4402:=
4321:=
4251:=
4134:,
4065:=
3967:=
3856:=
3792:≡
3753:=
3671:=
3589:−
3571:π
3554:=
3369:)
3316:χ
3307:−
3281:−
3265:−
3246:π
3226:−
3198:π
3189:−
3162:∞
3153:∫
3134:−
3111:¯
3108:χ
3100:∞
3085:∑
3067:−
3053:−
3044:δ
3032:χ
3023:∑
3017:∞
3008:∫
2996:−
2979:χ
2974:∞
2959:∑
2943:χ
2896:π
2876:−
2848:π
2839:−
2802:¯
2799:χ
2791:∞
2776:∑
2753:−
2744:δ
2732:χ
2723:∑
2690:¯
2687:χ
2616:χ
2591:π
2565:χ
2557:∈
2550:∑
2538:∈
2532:∀
2512:χ
2476:π
2453:χ
2445:∈
2438:∑
2394:−
2377:χ
2374:−
2356:∏
2327:χ
2315:∑
2290:∗
2280:∈
2215:−
2183:φ
2110:≡
2058:
2016:ω
2000:χ
1926:−
1912:π
1892:ω
1869:−
1843:×
1769:≡
1731:χ
1700:χ
1679:χ
1612:−
1473:−
1431:×
1317:J.
1295:)
1291:•
1262:)
1247:J.
1242:.—
1222:)
1195:J.
1149:)
1125:J.
1120:.—
1079:χ
1055:χ
999:∈
950:)
928:)
902:)
845:)
813:ω
809:−
788:χ
767:ω
764:−
752:ω
729:ω
697:χ
675:χ
648:χ
605:π
599:
590:ω
570:π
533:π
527:
518:ω
440:Hv
416:Hv
406:0
403:1
374:χ
312:ϕ
290:1
261:χ
202:ϕ
9092:(
9062:(
9038:(
9014:(
8977:(
8954:n
8931:)
8928:1
8919:(
8915:/
8910:i
8904:2
8899:e
8877:)
8869:3
8864:(
8835:)
8830:5
8827:a
8822:(
8817:)
8812:3
8809:a
8804:(
8799:)
8794:2
8791:a
8786:(
8764:)
8759:6
8756:a
8751:(
8746:)
8741:5
8738:a
8733:(
8711:)
8703:a
8698:(
8693:)
8688:3
8685:a
8680:(
8658:)
8650:a
8645:(
8640:)
8635:2
8632:a
8627:(
8605:)
8597:a
8592:(
8563:)
8560:1
8551:(
8547:/
8542:i
8536:2
8531:e
8509:)
8501:2
8496:(
8467:)
8462:7
8459:a
8454:(
8449:)
8444:4
8441:a
8436:(
8414:)
8406:a
8401:(
8372:)
8367:1
8361:3
8357:3
8350:)
8347:1
8341:3
8338:(
8335:(
8331:/
8326:i
8320:2
8315:e
8293:)
8285:2
8280:(
8269:a
8265:a
8243:)
8235:a
8230:(
8225:)
8220:2
8217:a
8212:(
8190:)
8182:a
8177:(
8148:)
8143:1
8137:2
8133:5
8126:)
8123:1
8117:5
8114:(
8111:(
8107:/
8102:i
8096:2
8091:e
8069:)
8061:2
8056:(
8045:a
8041:a
8019:)
8014:8
8011:a
8006:(
8001:)
7996:3
7993:a
7988:(
7966:)
7958:a
7953:(
7924:)
7921:1
7912:(
7908:/
7903:i
7897:2
7892:e
7870:)
7862:5
7857:(
7828:)
7820:a
7815:(
7810:)
7805:2
7802:a
7797:(
7775:)
7767:a
7762:(
7734:)
7729:7
7726:a
7721:(
7716:)
7711:3
7708:a
7703:(
7681:)
7673:a
7668:(
7640:)
7635:5
7632:a
7627:(
7622:)
7617:4
7614:a
7609:(
7587:)
7579:a
7574:(
7545:)
7542:1
7533:(
7529:/
7524:i
7518:2
7513:e
7491:)
7483:2
7478:(
7449:)
7444:9
7441:a
7436:(
7431:)
7426:2
7423:a
7418:(
7396:)
7388:a
7383:(
7354:)
7351:1
7342:(
7338:/
7333:i
7327:2
7322:e
7300:)
7292:3
7287:(
7228:n
7168:)
7163:5
7160:a
7155:(
7150:)
7145:3
7142:a
7137:(
7115:)
7107:a
7102:(
7040:)
7035:7
7032:a
7027:(
7022:)
7017:2
7014:a
7009:(
6987:)
6979:a
6974:(
6905:)
6902:1
6893:(
6889:/
6884:i
6878:2
6873:e
6851:)
6843:2
6838:(
6781:)
6776:4
6773:a
6768:(
6763:)
6758:3
6755:a
6750:(
6728:)
6720:a
6715:(
6652:)
6649:1
6640:(
6636:/
6631:i
6625:2
6620:e
6598:)
6590:2
6585:(
6532:)
6527:5
6524:a
6519:(
6514:)
6509:2
6506:a
6501:(
6479:)
6471:a
6466:(
6411:)
6406:1
6400:2
6396:3
6389:)
6386:1
6380:3
6377:(
6374:(
6370:/
6365:i
6359:2
6354:e
6332:)
6327:9
6324:2
6319:(
6308:a
6304:a
6261:)
6256:8
6253:a
6248:(
6234:n
6189:)
6186:1
6180:7
6177:(
6173:/
6168:i
6162:2
6157:e
6135:)
6130:7
6127:3
6122:(
6079:)
6074:3
6071:a
6066:(
6061:)
6056:2
6053:a
6048:(
6026:)
6021:6
6018:a
6013:(
5968:)
5965:1
5959:5
5956:(
5952:/
5947:i
5941:2
5936:e
5914:)
5909:5
5906:2
5901:(
5858:)
5853:1
5847:2
5843:2
5836:)
5833:1
5827:2
5824:(
5821:(
5817:/
5812:i
5806:2
5801:e
5779:)
5774:4
5771:3
5766:(
5755:a
5751:a
5747:n
5714:)
5711:1
5705:3
5702:(
5698:/
5693:i
5687:2
5682:e
5660:)
5655:3
5652:2
5647:(
5629:0
5610:)
5607:1
5601:2
5598:(
5594:/
5589:i
5583:2
5578:e
5556:)
5551:2
5548:1
5543:(
5525:0
5508:)
5503:1
5500:0
5495:(
5426:a
5422:n
5383:r
5363:)
5351:i
5348:+
5336:(
5333:r
5308:i
5304:e
5300:r
5280:]
5274:,
5271:r
5268:[
5258:n
5254:a
5239:)
5234:n
5231:a
5226:(
5212:n
5208:a
5192:k
5187:)
5182:n
5179:a
5174:(
5162:k
5148:)
5145:n
5142:(
5115:)
5112:n
5109:(
5085:)
5080:n
5077:a
5072:(
5061:n
5057:a
5017:)
5014:n
5011:(
5003:)
4998:n
4995:a
4990:(
4978:n
4974:a
4966:n
4951:)
4946:n
4943:a
4938:(
4927:a
4923:k
4908:)
4901:k
4897:2
4892:a
4884:k
4880:2
4873:(
4851:)
4844:k
4840:2
4836:a
4831:(
4818:.
4816:i
4801:)
4788:(
4766:)
4758:5
4753:(
4742:i
4727:)
4714:(
4692:)
4684:3
4679:(
4657:)
4649:9
4644:(
4622:)
4614:7
4609:(
4587:)
4574:(
4552:)
4544:1
4539:(
4524:p
4520:p
4516:a
4512:p
4508:a
4504:p
4500:p
4496:a
4492:a
4488:k
4484:p
4468:)
4463:1
4457:k
4453:p
4446:)
4443:1
4437:p
4434:(
4431:(
4427:/
4422:i
4416:2
4411:e
4389:)
4382:k
4378:p
4374:a
4369:(
4343:)
4338:8
4335:5
4330:(
4308:)
4303:8
4300:3
4295:(
4273:)
4268:8
4265:7
4260:(
4238:)
4233:8
4230:1
4225:(
4199:)
4194:4
4191:3
4186:(
4164:)
4159:4
4156:1
4151:(
4136:n
4132:m
4128:n
4124:m
4120:a
4105:)
4100:n
4097:a
4092:(
4087:)
4082:m
4079:a
4074:(
4052:)
4046:n
4043:m
4039:a
4034:(
4019:n
4015:k
4011:a
3995:k
3990:)
3985:n
3982:a
3977:(
3954:)
3949:n
3944:k
3940:a
3934:(
3919:n
3915:b
3911:a
3896:)
3891:n
3888:b
3883:(
3878:)
3873:n
3870:a
3865:(
3843:)
3838:n
3834:b
3831:a
3825:(
3810:n
3806:b
3802:a
3798:n
3794:b
3790:a
3775:)
3770:n
3767:b
3762:(
3740:)
3735:n
3732:a
3727:(
3716:n
3712:k
3708:a
3693:)
3688:n
3685:a
3680:(
3658:)
3653:n
3649:n
3646:k
3643:+
3640:a
3634:(
3621:.
3619:p
3615:a
3611:p
3595:)
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3586:p
3583:(
3579:/
3574:i
3568:2
3563:e
3541:)
3536:p
3533:a
3528:(
3513:n
3498:)
3493:n
3490:1
3485:(
3459:)
3454:1
3451:0
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3432:n
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3424:n
3420:a
3405:)
3400:n
3397:a
3392:(
3365:(
3331:)
3328:s
3325:(
3322:A
3319:)
3313:,
3310:s
3304:1
3301:(
3298:L
3295:=
3292:x
3289:d
3284:s
3277:x
3272:)
3268:2
3260:q
3256:/
3252:x
3249:k
3243:i
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3229:1
3223:(
3220:Y
3217:+
3212:q
3208:/
3204:x
3201:k
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3178:1
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3172:Y
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3144:1
3137:1
3131:s
3127:k
3123:)
3120:k
3117:(
3095:1
3092:=
3089:k
3081:=
3078:x
3075:d
3070:s
3063:x
3059:)
3056:n
3050:x
3047:(
3041:)
3038:n
3035:(
3027:n
3012:0
3004:=
2999:s
2992:n
2988:)
2985:n
2982:(
2969:1
2966:=
2963:n
2955:=
2952:)
2949:s
2946:,
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2937:L
2916:)
2910:q
2906:/
2902:x
2899:k
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2825:(
2822:Y
2818:(
2814:)
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2808:(
2786:1
2783:=
2780:k
2770:q
2767:1
2762:=
2759:)
2756:n
2750:x
2747:(
2741:)
2738:n
2735:(
2727:n
2702:)
2699:n
2696:(
2681:)
2678:1
2675:(
2672:Y
2669:=
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2663:k
2660:(
2657:Y
2637:)
2634:k
2631:(
2628:Y
2625:)
2622:a
2619:(
2613:=
2608:p
2604:/
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2597:n
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2588:i
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2448:G
2442:n
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2422:Y
2397:s
2390:p
2386:)
2383:p
2380:(
2371:1
2367:1
2360:p
2352:=
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2285:N
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2238:p
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2163:p
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2134:p
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2061:(
2053:g
2045:=
2042:a
2020:a
2012:=
2009:)
2006:n
2003:(
1979:)
1976:p
1968:(
1962:0
1959:≢
1956:n
1932:)
1929:1
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1920:(
1916:/
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1836:Z
1832:p
1828:/
1823:Z
1819:(
1816:=
1813:G
1802:,
1789:)
1786:p
1778:(
1772:0
1766:n
1746:0
1743:=
1740:)
1737:n
1734:(
1709:)
1706:n
1703:(
1697:=
1694:)
1691:p
1688:+
1685:n
1682:(
1659:p
1632:)
1629:+
1626:,
1622:Z
1618:)
1615:1
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1602:/
1597:Z
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1573:g
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1533:G
1513:g
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1467:(
1463:/
1458:Z
1454:(
1434:)
1428:,
1424:Z
1420:p
1416:/
1411:Z
1407:(
1404:=
1401:G
1381:p
1360:Z
1356:p
1352:/
1347:Z
1310:d
1306:d
1287:(
1258:(
1235:p
1231:p
1218:(
1188:Z
1185:n
1183:/
1181:Z
1177:s
1173:L
1169:s
1165:L
1145:(
1138:n
1136:/
1134:Z
1118:n
1114:n
1110:n
1106:n
1102:a
1088:)
1085:k
1082:(
1076:=
1073:)
1070:m
1067:a
1064:+
1061:k
1058:(
1045:m
1031:}
1028:1
1025:=
1022:)
1019:q
1016:,
1013:n
1010:(
1007::
1003:Z
996:q
993:{
983:Z
980:n
978:/
976:Z
972:k
970:,
968:n
964:k
960:n
946:(
924:(
918:Z
915:n
913:/
911:Z
898:(
884:q
880:n
876:q
874:,
872:n
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841:(
817:2
806:=
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800:2
797:(
792:2
761:=
756:4
748:=
743:2
739:)
733:2
725:(
722:=
717:2
713:)
709:3
706:(
701:2
693:=
690:)
687:2
684:(
679:2
652:2
619:)
616:3
612:/
608:i
602:(
593:=
567:2
547:)
544:6
540:/
536:i
530:(
521:=
389:)
386:n
383:(
378:1
360:2
355:1
350:n
327:1
324:=
321:)
318:2
315:(
276:)
273:n
270:(
265:1
247:1
242:n
217:1
214:=
211:)
208:1
205:(
154:.
53::
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