1667:. Il seroit peut-être nécessaire de démontrer rigoureusement une chose que nous avons supposée dans plusieurs endroits de cet article, savoir, qu'il y a une infinité de nombres premiers compris dans tous progression arithmétique, dont le premier terme & la raison sont premiers entr'eux, ou, ce qui revient au même, dans la formule 2mx + μ, lorsque 2m & μ n'ont point de commun diviseur. Cette proposition est assez difficile à démontrer, cependant on peut s'assurer qu'elle est vraie, en comparant la progression arithmétique dont il s'agit, à la progression ordinaire 1, 3, 5, 7, &c. Si on prend un grand nombre de termes de ces progressions, le même dans les deux, & qu'on les dispose, par exemple, de manière que le plus grand terme soit égal & à la même place de part & d'autre; on verra qu'en omettant de chaque côté les multiples de 3, 5, 7, &c. jusqu'à un certain nombre premier
1677:. It will perhaps be necessary to prove rigorously something that we have assumed at several places in this article, namely, that there is an infinitude of prime numbers included in every arithmetic progression, whose first term and common difference are co-prime, or, what amounts to the same thing, in the formula 2mx + μ, when 2m and μ have no common divisors at all. This proposition is rather difficult to prove, however one may be assured that it is true, by comparing the arithmetic progression being considered to the ordinary progression 1, 3, 5, 7, etc. If one takes a great number of terms of these progressions, the same in both, and if one arranges them, for example, in a way that the largest term be equal and at the same place in both; one will see that by omitting from each the multiples of 3, 5, 7, etc., up to a certain prime number
1625:− 1, as all of the first are sums of two squares, but the latter are thoroughly excluded from this property: reciprocal series formed from both classes, namely: 1/5 + 1/13 + 1/17 + 1/29 + etc. and 1/3 + 1/7 + 1/11 + 1/19 + 1/23 + etc. will both be equally infinite, which likewise is to be had from all types of prime numbers. Thus, if there be chosen from the prime numbers only those that are of the form 100n + 1, of which kind are 101, 401, 601, 701, etc., not only the set of these is infinite, but likewise the sum of the series formed from that , namely: 1/101 + 1/401 + 1/601 + 1/701 + 1/1201 + 1/1301 + 1/1601 + 1/1801 + 1/1901 + etc. likewise is infinite.)
1737:"Soit donnée une progression arithmétique quelconque A − C, 2A − C, 3A − C, etc., dans laquelle A et C sont premiers entre eux; soit donnée aussi une suite θ, λ, μ … ψ, ω, composée de k nombres premiers impairs, pris à volonté et disposés dans un order quelconque; si on appelle en général π le z terme de la suite naturelle des nombres premiers 3, 5, 7, 11, etc., je dis que sur π termes consécutifs de la progression proposée, il y en aura au moins un qui ne sera divisible par aucun des nombres premiers θ, λ, μ … ψ, ω."
2159:
1694:"XIX. … En général, a étant un nombre donné quelconque, tout nombres impair peut être représenté par la formule 4ax ± b, dans laquelle b est impair et moindre que 2a. Si parmi tous les valeurs possibles de b on retranche celles qui ont un commun diviseur avec a, les formes restantes 4ax ± b comprendront tous les nombres premiers partagé, … "
420:
1599:"Quoniam porro numeri primi praeter binarium quasi a natura in duas classes distinguuntur, prouti fuerint vel formae 4n + 1, vel formae 4n − 1, dum priores omnes sunt summae duorum quadratorum, posteriores vero ab hac proprietate penitus excluduntur: series reciprocae ex utraque classes formatae, scillicet:
1779:
term of the natural series of prime numbers 3, 5, 7, 11, etc., I claim that among the π consecutive terms of the proposed progression, there will be at least one of them that will not be divisible by any of the prime numbers θ, λ, μ … ψ, ω.) This assertion was proven false in 1858 by
Anthanase Louis
1608:
ambae erunt pariter infinitae, id quod etiam de omnibus speciebus numerorum primorum est tenendum. Ita si ex numeris primis ii tantum excerpantur, qui sunt formae 100n + 1, cuiusmodi sunt 101, 401, 601, 701, etc., non solum multitudo eorum est infinita, sed etiam summa huius seriei ex illis
1869:
denotes a variable, surely prime numbers are contained — seems difficult enough, and incidentally, he points out a method that could perhaps lead to it; however, many preliminary and necessary investigations are seen by us before this may indeed reach the path to a rigorous
197:
of the prime numbers in the progression diverges and that different such arithmetic progressions with the same modulus have approximately the same proportions of primes. Equivalently, the primes are evenly distributed (asymptotically) among the congruence classes modulo
1846:
indefinitum, certo contineri numeros primos, satis difficilem videri, methodumque obiter addigitat, quae forsan illuc conducere possit; multae vero disquisitiones praeliminares necessariae nobis videntur, antequam hacce quidem via ad demonstrationem rigorosam pervenire
275:
1671:, il doit rester des deux côtés le même nombre de termes, ou même il en restera moins dans la progression 1, 3, 5, 7, &c. Mais comme dans celle-ci, il reste nécessairement des nombres premiers, il en doit rester aussi dans l'autre."
1681:, there should remain in both the same number of terms, or even there will remain fewer of them in the progression 1, 3, 5, 7, etc. But as in this , there necessarily remain prime numbers, there shall also remain some in the other .)
1985:[Proof of the theorem that every unbounded arithmetic progression, whose first term and common difference are integers without common factors, contains infinitely many prime numbers],
184:
797:, the same must be true for the primes in arithmetic progressions. It is natural to ask about the way the primes are shared between the various arithmetic progressions for a given value of
1983:"Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält"
1171:
914:
2205:
1063:
997:
415:{\displaystyle {\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{19}}+{\frac {1}{23}}+{\frac {1}{31}}+{\frac {1}{43}}+{\frac {1}{47}}+{\frac {1}{59}}+{\frac {1}{67}}+\cdots }
1581:+ 1 signum negativum" (On the sum of series of prime numbers arranged 1/3 − 1/5 + 1/7 + 1/11 − 1/13 − 1/17 + 1/19 + 1/23 − 1/29 + 1/31 etc., where the prime numbers of the form 4
1221:
468: + 2 except for the only even prime 2. The following table lists several arithmetic progressions with infinitely many primes and the first few ones in each of them.
864:
1286:, and that the ratio is infinite. In 1775, Euler stated the theorem for the cases of a + nd, where a = 1. This special case of Dirichlet's theorem can be proven using
1272:
1087:
1233:
When compared to each other, progressions with a quadratic nonresidue remainder have typically slightly more elements than those with a quadratic residue remainder (
265:
0, 1, 2, 4, 5, 7, 10, 11, 14, 16, 17, 19, 20, 25, 26, 31, 32, 34, 37, 40, 41, 44, 47, 49, 52, 55, 56, 59, 62, 65, 67, 70, 76, 77, 82, 86, 89, 91, 94, 95, ...
1573:
Leonhard Euler, "De summa seriei ex numeris primis formatae 1/3 − 1/5 + 1/7 + 1/11 − 1/13 − 1/17 + 1/19 + 1/23 − 1/29 + 1/31 etc. ubi numeri primi formae 4
484:
258:
236:
243:
3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, ...
193:
that there are infinitely many prime numbers. Stronger forms of
Dirichlet's theorem state that for any such arithmetic progression, the sum of the
2191:
2286:
1923:
1478:
Shiu showed that any arithmetic progression satisfying the hypothesis of
Dirichlet's theorem will in fact contain arbitrarily long runs of
1506:
1617:(Since, further, prime numbers larger than two are divided as if by Nature into two classes, according as they were either of the form 4
2240:
2099:
2014:
2198:
1491:
1431:
1245:
In 1737, Euler related the study of prime numbers to what is known now as the
Riemann zeta function: he showed that the value
2214:
2177:
1978:
1309:
214:
2291:
2087:
1378:) can be proven by analyzing the splitting behavior of primes in cyclotomic extensions, without making use of calculus (
1496:
1394:
generalizes
Dirichlet's theorem to higher-degree polynomials. Whether or not even simple quadratic polynomials such as
103:
1304:
1501:
1440:(1944) concerns the size of the smallest prime in a given arithmetic progression. Linnik proved that the progression
826:
1420:
1093:
875:
444:
are often ignored because half the numbers are even and the other half is the same numbers as a sequence with 2
189:
and
Dirichlet's theorem states that this sequence contains infinitely many prime numbers. The theorem extends
2167:
There are infinitely many prime numbers in all arithmetic progressions with first term and difference coprime
1821:
1423:
generalizes these two conjectures, i.e. generalizes to more than one polynomial with degree larger than one.
1427:
1413:
1359:
1003:
937:
2225:
1391:
1363:
1329:
194:
95:
1511:
2255:
2045:
2003:
Algebraic number theory. Translated from the 1992 German original and with a note by
Norbert Schappacher
1325:
1295:
1291:
1287:
1982:
1179:
1658:
1650:
1402:
1355:
1299:
794:
788:
834:
2230:
1437:
1234:
190:
73:
1689:
17:
2265:
2141:
2062:
1317:
80:
2043:(1949), "An elementary proof of Dirichlet's theorem about primes in an arithmetic progression",
1998:
1248:
2260:
2235:
2095:
2010:
1949:
1919:
1732:
1594:
1562:
1546:
1532:
1069:
1801:
2245:
2133:
2117:
2105:
2070:
2054:
2028:
1937:
1889:
1788:
1655:
Histoire de l'Académie royale des sciences, avec les mémoires de mathématique et de physique
1472:
1343:
426:
2024:
1933:
2109:
2091:
2074:
2032:
2020:
2006:
1941:
1929:
1775:
odd prime numbers, taken at will and arranged in any order; if one calls in general π the
1771:
are prime among themselves ; let there be given also a series θ, λ, μ … ψ, ω composed of
1593:(St. Petersburg, Russia: Imperial Academy of Sciences, 1785), vol. 2, pp. 240–256;
2173:
1952:
1911:
1849:(The illustrious Le Gendre himself admits the proof of the theorem — among the form
72:
is also a positive integer. In other words, there are infinitely many primes that are
2280:
31:
2145:
809:
their terms). The answer is given in this form: the number of feasible progressions
2121:
2040:
1406:
1335:
57:
39:
2183:
1612:
1/101 + 1/401 + 1/601 + 1/701 + 1/1201 + 1/1301 + 1/1601 + 1/1801 + 1/1901 + etc.
1971:
1471:
An analogue of
Dirichlet's theorem holds in the framework of dynamical systems (
1328:
to the distribution of primes. The theorem represents the beginning of rigorous
1987:
Abhandlungen der Königlichen Preußischen
Akademie der Wissenschaften zu Berlin
1893:
806:
1967:
1957:
1918:, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag,
1452:
ranges through the positive integers) contains a prime of magnitude at most
789:
Prime number theorem § Prime number theorem for arithmetic progressions
1798:
The
Development of Prime Number Theory: From Euclid to Hardy and Littlewood
1826:"Ill. Le Gendre ipse fatetur, demonstrationem theorematis, sub tali forma
1700:
being any given number, all odd numbers can be represented by the formula
1362:) at 1 is nonzero. The proof of this statement requires some calculus and
2005:, Grundlehren der Mathematischen Wissenschaften , vol. 322, Berlin:
805:
of those, essentially, if we do not distinguish two progressions sharing
2137:
2066:
46:
43:
1790:
Examen d'une proposition de Legendre relative à la théorie des nombres
1374:= 1 (i.e., concerning the primes that are congruent to 1 modulo some
2058:
1561:(Washington, D.C.: The Mathematical Association of America, 2007),
2164:
1820:(Leipzig, (Germany): Gerhard Fleischer, Jr., 1801), Section 297,
2187:
1354:
Dirichlet's theorem is proved by showing that the value of the
793:
Since the primes thin out, on average, in accordance with the
1416:
generalizes Dirichlet's theorem to more than one polynomial.
1324:. The proof is modeled on Euler's earlier work relating the
1537:
Commentarii Academiae Scientiarum Imperialis Petropolitanae
1290:. The general form of the theorem was first conjectured by
825:
do not have a common factor > 1 — is given by
774:
757:
740:
723:
706:
689:
672:
655:
638:
621:
604:
587:
570:
553:
536:
519:
502:
253:
231:
1968:"Dirichlet's Theorem on Primes in Arithmetic Progressions"
27:
Theorem on the number of primes in arithmetic sequences
2170:
English translation of the original paper at the arXiv
1880:
Shiu, D. K. L. (2000). "Strings of congruent primes".
1535:[Various observations about infinite series].
1405:) attain infinitely many prime values is an important
869:
Further, the proportion of primes in each of those is
1251:
1182:
1096:
1072:
1006:
940:
878:
837:
278:
106:
269:
The strong form of Dirichlet's theorem implies that
213:
The theorem is named after the German mathematician
1716:one removes those that have a common divisor with
1266:
1215:
1165:
1081:
1057:
991:
908:
858:
414:
178:
179:{\displaystyle a,\ a+d,\ a+2d,\ a+3d,\ \dots ,\ }
1865:denote given integers prime among themselves
1585:− 1 have a positive sign, whereas of the form 4
720:13, 37, 61, 73, 97, 109, 157, 181, 193, 229, ...
1739:(Let there be given any arithmetic progression
1653:(Investigations of interdeterminate analysis),
1274:reduces to a ratio of two infinite products, Π
771:11, 23, 47, 59, 71, 83, 107, 131, 167, 179, ...
2251:Dirichlet's theorem on arithmetic progressions
1688:(Paris, France: Duprat, 1798), Introduction,
754:7, 19, 31, 43, 67, 79, 103, 127, 139, 151, ...
737:5, 17, 29, 41, 53, 89, 101, 113, 137, 149, ...
703:19, 29, 59, 79, 89, 109, 139, 149, 179, 199, …
652:11, 31, 41, 61, 71, 101, 131, 151, 181, 191, …
584:17, 41, 73, 89, 97, 113, 137, 193, 233, 241, …
2199:
2090:, vol. 7, New York; Heidelberg; Berlin:
1533:"Variae observationes circa series infinitas"
8:
1731:, 2nd ed. (Paris, France: Courcier, 1808),
1166:{\displaystyle q+q-1,2q+q-1,3q+q-1,\dots \ }
686:7, 17, 37, 47, 67, 97, 107, 127, 137, 157, …
635:7, 23, 31, 47, 71, 79, 103, 127, 151, 167, …
618:5, 13, 29, 37, 53, 61, 101, 109, 149, 157, …
456: + 1 produces the same primes as 3
1589:+ 1 a negative sign.) in: Leonhard Euler,
1577:− 1 habent signum positivum, formae autem 4
909:{\displaystyle {\frac {1}{\varphi (d)}}.\ }
669:3, 13, 23, 43, 53, 73, 83, 103, 113, 163, …
601:3, 11, 19, 43, 59, 67, 83, 107, 131, 139, …
247:They correspond to the following values of
2206:
2192:
2184:
42:theorem, states that for any two positive
1793:(Paris, France: Mallet-Bachelier, 1859).
1313:
1250:
1181:
1095:
1071:
1005:
939:
879:
877:
836:
396:
383:
370:
357:
344:
331:
318:
305:
292:
279:
277:
105:
1724:include all prime numbers among them … )
1634:
1379:
1294:in his attempted unsuccessful proofs of
567:5, 11, 17, 23, 29, 41, 47, 53, 59, 71, …
550:7, 13, 19, 31, 37, 43, 61, 67, 73, 79, …
516:5, 13, 17, 29, 37, 41, 53, 61, 73, 89, …
470:
1559:The Early Mathematics of Leonhard Euler
1523:
1339:
533:3, 7, 11, 19, 23, 31, 43, 47, 59, 67, …
1916:Introduction to analytic number theory
1464:. Subsequent researchers have reduced
1058:{\displaystyle q+2,2q+2,3q+2,\dots \ }
992:{\displaystyle q+1,2q+1,3q+1,\dots \ }
499:3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …
2160:Scans of the original paper in German
1605:1/3 + 1/7 + 1/11 + 1/19 + 1/23 + etc.
1430:, Dirichlet's theorem generalizes to
1367:
464: + 5 produces the same as 3
7:
1800:(Berlin, Germany: Springer, 2000);
2122:"Closed orbits in homology classes"
1712:. If among all possible values of
1651:"Recherches d'analyse indéterminée"
2241:Dirichlet-multinomial distribution
1602:1/5 + 1/13 + 1/17 + 1/29 + etc. et
481:First 10 of infinitely many primes
25:
18:Dirichlet's theorem on primes
1507:Dirichlet's approximation theorem
1729:Essai sur la Théorie des Nombres
1686:Essai sur la Théorie des Nombres
1216:{\displaystyle q,2q,3q,\dots \ }
1842:numeros inter se primos datos,
1230: − 1) of the primes.
2215:Peter Gustav Lejeune Dirichlet
2178:Wolfram Demonstrations Project
1261:
1255:
894:
888:
859:{\displaystyle \varphi (d).\ }
847:
841:
452: = 0. For example, 6
215:Peter Gustav Lejeune Dirichlet
1:
2088:Graduate Texts in Mathematics
2287:Theorems about prime numbers
1432:Chebotarev's density theorem
931: − 1 progressions
56:, there are infinitely many
38:, also called the Dirichlet
2120:; Katsuda, Atsushi (1990),
2082:Serre, Jean-Pierre (1973),
1818:Disquisitiones arithmeticae
1659:see especially p. 552.
1492:Bombieri–Vinogradov theorem
1305:Disquisitiones Arithmeticae
2308:
1802:see especially p. 50.
1547:Theorema 7 on pp. 172–174.
786:
86:. The numbers of the form
2221:
1894:10.1112/s0024610799007863
1720:, the remaining formulas
1267:{\displaystyle \zeta (1)}
1226:contains a proportion 1/(
217:, who proved it in 1837.
1780:Dupré (1808–1869). See:
1531:Euler, Leonhard (1737).
1082:{\displaystyle \dots \ }
827:Euler's totient function
225:The primes of the form 4
1796:Narkiewicz, Władysław,
1497:Brun–Titchmarsh theorem
1475:and A. Katsuda, 1990).
1456:for absolute constants
1428:algebraic number theory
1421:Schinzel's hypothesis H
1403:Landau's fourth problem
1370:). The particular case
1308:— but it was proved by
460: + 1, while 6
2226:Dirichlet distribution
2084:A course in arithmetic
1816:Carl Friedrich Gauss,
1502:Siegel–Walfisz theorem
1392:Bunyakovsky conjecture
1364:analytic number theory
1330:analytic number theory
1288:cyclotomic polynomials
1268:
1217:
1167:
1083:
1059:
993:
910:
860:
416:
180:
96:arithmetic progression
2256:Dirichlet convolution
2046:Annals of Mathematics
1953:"Dirichlet's Theorem"
1708:is odd and less than
1621:+ 1, or of the form 4
1557:Sandifer, C. Edward,
1326:Riemann zeta function
1296:quadratic reciprocity
1269:
1218:
1168:
1084:
1060:
994:
911:
861:
417:
181:
2292:Zeta and L-functions
1696:(XIX. … In general,
1657:, pp. 465–559;
1637:, §I.10, Exercise 1.
1615:etiam est infinita."
1609:formatae, scillicet:
1414:Dickson's conjecture
1356:Dirichlet L-function
1282:–1), for all primes
1249:
1180:
1094:
1070:
1004:
938:
876:
835:
795:prime number theorem
276:
104:
2231:Dirichlet character
2176:by Jay Warendorff,
2174:Dirichlet's Theorem
1979:Dirichlet, P. G. L.
1882:J. London Math. Soc
1824:From pp. 507–508:
1735:From p. 404:
1661:From p. 552:
1597:From p. 241:
472:
448:, if we start with
36:Dirichlet's theorem
2266:Dirichlet integral
2138:10.1007/BF02699875
1950:Weisstein, Eric W.
1692:From p. 12:
1591:Opuscula analytica
1358:(of a non-trivial
1264:
1213:
1163:
1079:
1055:
989:
923:is a prime number
906:
856:
471:
412:
229:+ 3 are (sequence
176:
2274:
2273:
2261:Dirichlet problem
2236:Dirichlet process
2118:Sunada, Toshikazu
1925:978-0-387-90163-3
1787:Dupré, A. (1859)
1763:, etc., in which
1649:Le Gendre (1785)
1512:Green–Tao theorem
1212:
1162:
1078:
1054:
988:
905:
898:
855:
780:
779:
404:
391:
378:
365:
352:
339:
326:
313:
300:
287:
175:
166:
148:
130:
115:
16:(Redirected from
2299:
2246:Dirichlet series
2208:
2201:
2194:
2185:
2148:
2126:Publ. Math. IHÉS
2112:
2077:
2035:
1999:Neukirch, Jürgen
1994:
1966:Chris Caldwell,
1963:
1962:
1944:
1898:
1897:
1877:
1871:
1834:, designantibus
1814:
1808:
1727:A. M. Legendre,
1684:A. M. Legendre,
1644:
1638:
1632:
1626:
1595:see p. 241.
1571:
1565:
1555:
1549:
1545:; specifically,
1544:
1528:
1438:Linnik's theorem
1400:
1344:elementary proof
1336:Atle Selberg
1273:
1271:
1270:
1265:
1235:Chebyshev's bias
1222:
1220:
1219:
1214:
1210:
1172:
1170:
1169:
1164:
1160:
1088:
1086:
1085:
1080:
1076:
1064:
1062:
1061:
1056:
1052:
998:
996:
995:
990:
986:
919:For example, if
915:
913:
912:
907:
903:
899:
897:
880:
865:
863:
862:
857:
853:
473:
427:divergent series
421:
419:
418:
413:
405:
397:
392:
384:
379:
371:
366:
358:
353:
345:
340:
332:
327:
319:
314:
306:
301:
293:
288:
280:
256:
234:
191:Euclid's theorem
185:
183:
182:
177:
173:
164:
146:
128:
113:
21:
2307:
2306:
2302:
2301:
2300:
2298:
2297:
2296:
2277:
2276:
2275:
2270:
2217:
2212:
2156:
2116:
2102:
2092:Springer-Verlag
2081:
2059:10.2307/1969454
2039:
2017:
2007:Springer-Verlag
1997:
1977:
1948:
1947:
1926:
1912:Apostol, Tom M.
1910:
1907:
1902:
1901:
1879:
1878:
1874:
1815:
1811:
1645:
1641:
1635:Neukirch (1999)
1633:
1629:
1572:
1568:
1556:
1552:
1530:
1529:
1525:
1520:
1488:
1482:prime numbers.
1395:
1388:
1386:Generalizations
1352:
1247:
1246:
1243:
1178:
1177:
1092:
1091:
1068:
1067:
1002:
1001:
936:
935:
884:
874:
873:
833:
832:
791:
785:
477:
274:
273:
252:
230:
223:
102:
101:
28:
23:
22:
15:
12:
11:
5:
2305:
2303:
2295:
2294:
2289:
2279:
2278:
2272:
2271:
2269:
2268:
2263:
2258:
2253:
2248:
2243:
2238:
2233:
2228:
2222:
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2154:External links
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2100:
2079:
2053:(2): 297–304,
2037:
2015:
1995:
1975:
1964:
1945:
1924:
1906:
1903:
1900:
1899:
1888:(2): 359–373.
1872:
1809:
1807:
1806:
1805:
1804:
1794:
1782:
1781:
1725:
1690:pp. 9–16.
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952:
949:
946:
943:
927:, each of the
917:
916:
902:
896:
893:
890:
887:
883:
867:
866:
852:
849:
846:
843:
840:
817:— those where
784:
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2135:
2131:
2127:
2123:
2119:
2115:
2111:
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2103:
2101:3-540-90040-3
2097:
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2041:Selberg, Atle
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2018:
2016:3-540-65399-6
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1380:Neukirch 1999
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1306:
1302:noted in his
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486:
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260:
255:
250:
242:
241:
240:
238:
233:
228:
220:
218:
216:
211:
209:
206:s coprime to
205:
201:
196:
192:
170:
167:
161:
158:
155:
152:
149:
143:
140:
137:
134:
131:
125:
122:
119:
116:
110:
107:
100:
99:
98:
97:
93:
90: +
89:
85:
82:
79:
75:
71:
67:
64: +
63:
59:
55:
51:
48:
45:
41:
37:
33:
32:number theory
19:
2250:
2166:
2129:
2125:
2083:
2050:
2044:
2002:
1990:
1986:
1956:
1915:
1885:
1881:
1875:
1866:
1862:
1858:
1854:
1850:
1845:
1841:
1837:
1833:
1829:
1825:
1822:pp. 507–508.
1817:
1812:
1797:
1789:
1776:
1772:
1768:
1764:
1760:
1756:
1752:
1748:
1744:
1740:
1736:
1733:p. 404.
1728:
1721:
1717:
1713:
1709:
1705:
1701:
1697:
1693:
1685:
1678:
1674:
1670:
1666:
1662:
1654:
1642:
1630:
1622:
1618:
1614:
1611:
1607:
1604:
1601:
1598:
1590:
1586:
1582:
1578:
1574:
1569:
1563:p. 253.
1558:
1553:
1540:
1536:
1526:
1479:
1477:
1470:
1465:
1461:
1457:
1453:
1449:
1445:
1441:
1436:
1425:
1418:
1411:
1407:open problem
1401:(known from
1396:
1389:
1375:
1371:
1353:
1334:
1319:
1303:
1283:
1279:
1275:
1244:
1232:
1227:
1225:
1176:(all except
1175:
928:
924:
920:
918:
868:
822:
818:
814:
810:
802:
798:
792:
783:Distribution
765:
748:
731:
714:
697:
680:
663:
646:
629:
612:
595:
578:
561:
544:
527:
510:
493:
465:
461:
457:
453:
449:
445:
441:
437:
433:
431:
424:
268:
251:: (sequence
248:
246:
226:
224:
212:
207:
203:
199:
188:
91:
87:
83:
77:
69:
65:
61:
60:of the form
53:
49:
40:prime number
35:
29:
2165:Dirichlet:
1972:Prime Pages
1704:, in which
1480:consecutive
1382:, §VII.6).
801:(there are
478:progression
202:containing
195:reciprocals
2281:Categories
2110:0256.12001
2075:0036.30603
2033:0956.11021
1942:0335.10001
1905:References
1543:: 160–188.
1368:Serre 1973
1342:) gave an
1318:Dirichlet
807:almost all
787:See also:
476:Arithmetic
432:Sequences
1958:MathWorld
1473:T. Sunada
1360:character
1310:Dirichlet
1253:ζ
1208:…
1158:…
1149:−
1128:−
1107:−
1074:…
1050:…
984:…
886:φ
839:φ
487:sequence
440:with odd
410:⋯
168:…
74:congruent
52:and
2146:26251216
2132:: 5–32,
2001:(1999),
1981:(1837),
1914:(1976),
1847:liceat."
1665:Remarque
1486:See also
1292:Legendre
221:Examples
94:form an
68:, where
47:integers
2067:1969454
2025:1697859
1993:: 45–71
1970:at the
1934:0434929
1870:proof.)
1722:4ax ± b
1702:4ax ± b
1338: (
1322:-series
1316:) with
1312: (
1241:History
775:A068231
758:A068229
741:A040117
724:A068228
707:A030433
690:A030432
673:A030431
656:A030430
639:A007522
622:A007521
605:A007520
588:A007519
571:A007528
554:A002476
537:A002145
520:A002144
503:A065091
257:in the
254:A095278
235:in the
232:A002145
44:coprime
2144:
2108:
2098:
2073:
2065:
2031:
2023:
2013:
1940:
1932:
1922:
1675:Remark
1663:"34.
1468:to 5.
1211:
1161:
1077:
1053:
987:
904:
854:
813:
811:modulo
174:
165:
147:
129:
114:
81:modulo
58:primes
2142:S2CID
2063:JSTOR
1673:(34.
1646:See:
1518:Notes
1350:Proof
1300:Gauss
1298:— as
1278:/ Π (
425:is a
2096:ISBN
2011:ISBN
1920:ISBN
1861:and
1767:and
1460:and
1448:(as
1419:The
1412:The
1390:The
1340:1949
1314:1837
821:and
768:+ 11
485:OEIS
259:OEIS
237:OEIS
2134:doi
2106:Zbl
2071:Zbl
2055:doi
2029:Zbl
1938:Zbl
1890:doi
1857:,
1755:, 3
1747:, 2
1426:In
1399:+ 1
1237:).
751:+ 7
734:+ 5
717:+ 1
700:+ 9
683:+ 7
666:+ 3
649:+ 1
632:+ 7
615:+ 5
598:+ 3
581:+ 1
564:+ 5
547:+ 1
530:+ 3
513:+ 1
496:+ 1
210:.
76:to
30:In
2283::
2140:,
2130:71
2128:,
2124:,
2104:,
2094:,
2086:,
2069:,
2061:,
2051:50
2049:,
2027:,
2021:MR
2019:,
2009:,
1991:48
1989:,
1955:.
1936:,
1930:MR
1928:,
1886:61
1884:.
1853:+
1851:kt
1838:,
1830:+
1828:kt
1759:−
1751:−
1743:−
1710:2a
1539:.
1454:cd
1446:nd
1434:.
1409:.
1346:.
1332:.
1223:)
764:12
747:12
730:12
713:12
696:10
679:10
662:10
645:10
434:dn
429:.
402:67
389:59
376:47
363:43
350:31
337:23
324:19
311:11
261:)
239:)
204:a'
92:nd
66:nd
34:,
2207:e
2200:t
2193:v
2180:.
2149:.
2136::
2113:.
2078:.
2057::
2036:.
1974:.
1961:.
1896:.
1892::
1867:t
1863:l
1859:k
1855:l
1844:t
1840:l
1836:k
1832:l
1777:z
1773:k
1769:C
1765:A
1761:C
1757:A
1753:C
1749:A
1745:C
1741:A
1718:a
1714:b
1706:b
1698:a
1679:p
1669:p
1623:n
1619:n
1587:n
1583:n
1579:n
1575:n
1541:9
1466:L
1462:L
1458:c
1450:n
1442:a
1397:x
1376:n
1372:a
1366:(
1320:L
1284:p
1280:p
1276:p
1262:)
1259:1
1256:(
1228:q
1205:,
1202:q
1199:3
1196:,
1193:q
1190:2
1187:,
1184:q
1155:,
1152:1
1146:q
1143:+
1140:q
1137:3
1134:,
1131:1
1125:q
1122:+
1119:q
1116:2
1113:,
1110:1
1104:q
1101:+
1098:q
1047:,
1044:2
1041:+
1038:q
1035:3
1032:,
1029:2
1026:+
1023:q
1020:2
1017:,
1014:2
1011:+
1008:q
981:,
978:1
975:+
972:q
969:3
966:,
963:1
960:+
957:q
954:2
951:,
948:1
945:+
942:q
929:q
925:q
921:d
901:.
895:)
892:d
889:(
882:1
851:.
848:)
845:d
842:(
823:d
819:a
815:d
803:d
799:d
766:n
749:n
732:n
715:n
698:n
681:n
664:n
647:n
630:n
628:8
613:n
611:8
596:n
594:8
579:n
577:8
562:n
560:6
545:n
543:6
528:n
526:4
511:n
509:4
494:n
492:2
466:n
462:n
458:n
454:n
450:n
446:d
442:d
438:a
407:+
399:1
394:+
386:1
381:+
373:1
368:+
360:1
355:+
347:1
342:+
334:1
329:+
321:1
316:+
308:1
303:+
298:7
295:1
290:+
285:3
282:1
249:n
227:n
208:d
200:d
171:,
162:,
159:d
156:3
153:+
150:a
144:,
141:d
138:2
135:+
132:a
126:,
123:d
120:+
117:a
111:,
108:a
88:a
84:d
78:a
70:n
62:a
54:d
50:a
20:)
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