2113:
2085:
1776:
has been translated from Gauss's
Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
1494:
614:
1653: : 74–104. Euler's theorem appears as "Theorema 11" on page 102. This paper was first presented to the Berlin Academy on June 8, 1758 and to the St. Petersburg Academy on October 15, 1759. In this paper, Euler's totient function,
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474:
1566:
219:
801:
857:
469:
1712:
385:
103:
420:
1680:
726:
240:
is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where
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1980:
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1489:{\displaystyle \prod _{i=1}^{\varphi (n)}x_{i}\equiv \prod _{i=1}^{\varphi (n)}ax_{i}=a^{\varphi (n)}\prod _{i=1}^{\varphi (n)}x_{i}{\pmod {n}},}
1995:
1839:
1709:
2020:
926:
2030:
2138:
1861:
1813:
1795:
609:{\displaystyle 7^{222}\equiv 7^{4\times 55+2}\equiv (7^{4})^{55}\times 7^{2}\equiv 1^{55}\times 7^{2}\equiv 49\equiv 9{\pmod {10}}}
1853:
2040:
2065:
2010:
1975:
17:
1806:
Untersuchungen uber hohere
Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
1510:
163:
1928:
1985:
1965:
1955:
2133:
1772:
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1990:
154:
755:
2103:
229:
1970:
806:
1960:
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2143:
2089:
1921:
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2060:
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347:
1294:), are identical (as sets—they may be listed in different orders), so the product of all the numbers in
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911:
72:
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2015:
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2005:
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The converse of Euler's theorem is also true: if the above congruence is true, then
2050:
930:
876:
236:
without proof), which is the restriction of Euler's theorem to the case where
1708:
For a review of Euler's work over the years leading to Euler's theorem, see:
1904:
1895:
1135:{\displaystyle a^{\varphi (n)}=a^{kM}=(a^{k})^{M}\equiv 1^{M}=1{\pmod {n}}.}
30:
This article is about Euler's theorem in number theory. For other uses, see
1782:
Gauss, Carl
Friedrich; Clarke, Arthur A. (translated into English) (1986),
1623:
For further details on this paper, including an
English translation, see:
868:
949:
66:
871:
communications. In this cryptosystem, Euler's theorem is used with
1804:
Gauss, Carl
Friedrich; Maser, H. (translated into German) (1965),
1610:"Theorematum quorundam ad numeros primos spectantium demonstratio"
1199:. The proof hinges on the fundamental fact that multiplication by
1913:
1850:
A Classical
Introduction to Modern Number Theory (Second edition)
317:. For example, consider finding the ones place decimal digit of
1917:
1690:; that is, the number of natural numbers that are smaller than
879:, and the security of the system is based on the difficulty of
1710:
Ed
Sandifer (2005) "Euler's proof of Fermat's little theorem"
1608:
Leonhard Euler (presented: August 2, 1736; published: 1741)
297:
The theorem may be used to easily reduce large powers modulo
18:
Proof of Euler-Fermat theorem using
Lagrange's theorem
1908:
891:
1. Euler's theorem can be proven using concepts from the
32:
List of things named after
Leonhard Euler § Theorems
1826:
1784:
Disquisitiones Arithemeticae (Second, corrected edition)
1612:(A proof of certain theorems regarding prime numbers),
1682:, is not named but referred to as "numerus partium ad
978:
form a subgroup of the group of residue classes, with
2101:
1659:
1647:
Novi Commentarii academiae scientiarum Petropolitanae
1645:(Proof of a new method in the theory of arithmetic),
1513:
1318:
1241:. (This law of cancellation is proved in the article
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divides the order of the entire group, in this case
1561:{\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}}.}
960:is in one of these residue classes, and its powers
903:form a group under multiplication (see the article
214:{\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}}.}
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1643:"Theoremata arithmetica nova methodo demonstrata"
1614:Commentarii academiae scientiarum Petropolitanae
290:The theorem is further generalized by some of
1496:and using the cancellation law to cancel each
796:{\displaystyle x\equiv y{\pmod {\varphi (n)}}}
1929:
8:
1287:, considered as sets of congruence classes (
852:{\displaystyle a^{x}\equiv a^{y}{\pmod {n}}}
929:states that the order of any subgroup of a
1936:
1922:
1914:
1729:Ireland & Rosen, corr. 1 to prop 3.3.2
1848:Ireland, Kenneth; Rosen, Michael (1990),
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464:{\displaystyle 7^{4}\equiv 1{\pmod {10}}}
445:
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387:. The integers 7 and 10 are coprime, and
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1701:For further details on this paper, see:
1686:primarum" (the number of parts prime to
1243:Multiplicative group of integers modulo
905:Multiplicative group of integers modulo
2108:
1598:
1305:) to the product of all the numbers in
699:are coprime), one needs to work modulo
1145:2. There is also a direct proof: Let
619:In general, when reducing a power of
7:
1822:Hardy, G. H.; Wright, E. M. (1980),
1547:
1475:
1121:
841:
776:
598:
453:
380:{\displaystyle 7^{222}{\pmod {10}}}
369:
200:
1996:Euler's continued fraction formula
25:
2021:Euler's pump and turbine equation
27:Theorem on modular exponentiation
2111:
2084:
2083:
2041:Euler equations (fluid dynamics)
2031:Euler's sum of powers conjecture
1540:
1468:
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834:
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98:{\displaystyle a^{\varphi (n)}}
1981:Euler–Poisson–Darboux equation
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895:: The residue classes modulo
863:Euler's theorem underlies the
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415:{\displaystyle \varphi (10)=4}
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84:
1:
875:being a product of two large
2011:Euler's four-square identity
422:. So Euler's theorem yields
2066:Euler–Bernoulli beam theory
1773:Disquisitiones Arithmeticae
1738:Hardy & Wright, thm. 72
1675:{\displaystyle \varphi (n)}
1641:L. Euler (published: 1763)
1004:, i.e. there is an integer
721:{\displaystyle \varphi (n)}
2160:
1195:be any integer coprime to
989:. Lagrange's theorem says
867:, which is widely used in
29:
2139:Theorems in number theory
2079:
1976:Euler–Mascheroni constant
1951:
1891:"Euler's Totient Theorem"
2026:Euler's rotation theorem
1872:Elementary Number Theory
1694:and relatively prime to
155:Euler's totient function
146:{\displaystyle \varphi }
69:positive integers, then
1986:Euler–Rodrigues formula
1966:Euler–Maclaurin formula
1956:Euler–Lagrange equation
1870:Landau, Edmund (1966),
1832:Oxford University Press
337:{\displaystyle 7^{222}}
230:Fermat's little theorem
51:Euler's totient theorem
2061:Euler number (physics)
1991:Euler–Tricomi equation
1676:
1562:
1503:gives Euler's theorem:
1490:
1456:
1391:
1348:
1182:reduced residue system
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2001:Euler's critical load
1971:Euler–Maruyama method
1808:, New York: Chelsea,
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1563:
1491:
1427:
1362:
1319:
1248:.) That is, the sets
1137:
1023:. This then implies,
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466:
417:
382:
339:
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292:Carmichael's theorems
282:
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228:published a proof of
216:
148:
120:
100:
1961:Euler–Lotka equation
1905:Euler-Fermat Theorem
1657:
1511:
1316:
1210:: in other words if
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899:that are coprime to
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47:Fermat–Euler theorem
1874:, New York: Chelsea
728:in the exponent of
45:(also known as the
2134:Modular arithmetic
1888:Weisstein, Eric W.
1715:2006-08-28 at the
1672:
1558:
1486:
1132:
927:Lagrange's theorem
910:for details). The
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53:) states that, if
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1841:978-0-19-853171-5
1703:The Euler Archive
1625:The Euler Archive
1583:Euler's criterion
1578:Carmichael number
914:of that group is
883:such an integer.
741:{\displaystyle a}
692:{\displaystyle n}
672:{\displaystyle a}
652:{\displaystyle n}
632:{\displaystyle a}
310:{\displaystyle n}
287:must be coprime.
280:{\displaystyle n}
260:{\displaystyle a}
118:{\displaystyle 1}
16:(Redirected from
2151:
2116:
2115:
2107:
2087:
2086:
2016:Euler's identity
1938:
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1588:Wilson's theorem
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893:theory of groups
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865:RSA cryptosystem
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832:
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2036:Euler's theorem
2006:Euler's formula
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1747:Landau, thm. 75
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1717:Wayback Machine
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1027:
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1005:
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990:
983:≡ 1 (mod
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957:
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945:
934:
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872:
823:
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43:Euler's theorem
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2144:Leonhard Euler
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1945:Leonhard Euler
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1880:External links
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1758:BĂ©zout's lemma
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1298:is congruent (
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1043:
1040:
1036:
948:is any number
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813:
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256:
244:is not prime.
226:Leonhard Euler
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2056:Euler numbers
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1903:
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1889:
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1879:
1873:
1868:
1865:
1863:0-387-97329-X
1859:
1855:
1851:
1846:
1843:
1837:
1833:
1828:
1827:
1820:
1817:
1815:0-8284-0191-8
1811:
1807:
1802:
1799:
1797:0-387-96254-9
1793:
1789:
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1780:
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1660:
1652:
1648:
1644:
1640:
1639:
1635:
1632:
1626:
1622:
1619:
1615:
1611:
1607:
1606:
1602:
1599:
1593:
1589:
1586:
1584:
1581:
1579:
1576:
1575:
1571:
1555:
1548:
1544:
1536:
1533:
1525:
1519:
1515:
1507:
1506:
1483:
1476:
1472:
1462:
1458:
1449:
1443:
1438:
1435:
1432:
1428:
1419:
1413:
1409:
1405:
1400:
1396:
1392:
1384:
1378:
1373:
1370:
1367:
1363:
1359:
1354:
1350:
1341:
1335:
1330:
1327:
1324:
1320:
1312:
1311:
1310:
1303:
1292:
1282:
1278:
1274:
1267:
1260:
1256:
1247:
1246:
1239:
1235:
1228:
1224:
1217:
1203:permutes the
1189:
1183:
1175:
1171:
1167:
1160:
1153:
1149:
1129:
1122:
1118:
1110:
1107:
1102:
1098:
1094:
1089:
1079:
1075:
1068:
1063:
1060:
1056:
1052:
1044:
1038:
1034:
1026:
1025:
1024:
1020:
1016:
1012:
1001:
997:
986:
982:
972:
968:
964:
951:
941:
937:
932:
928:
922:
918:
913:
909:
908:
894:
886:
884:
882:
878:
877:prime numbers
870:
866:
842:
838:
828:
824:
820:
815:
811:
783:
777:
773:
765:
762:
759:
751:
750:
749:
735:
712:
706:
686:
666:
646:
626:
617:
599:
595:
587:
584:
581:
578:
573:
569:
565:
560:
556:
552:
547:
543:
539:
534:
524:
520:
513:
508:
505:
502:
499:
496:
492:
488:
483:
479:
471:, and we get
454:
450:
442:
439:
434:
430:
409:
406:
400:
394:
370:
366:
356:
352:
329:
325:
304:
295:
293:
288:
274:
254:
245:
235:
231:
227:
208:
201:
197:
189:
186:
178:
172:
168:
160:
159:
158:
156:
140:
131:
127:
112:
87:
81:
77:
68:
63:
57:
52:
48:
44:
40:
39:number theory
33:
19:
2088:
2051:Euler method
2035:
1894:
1871:
1852:, New York:
1849:
1825:
1805:
1786:, New York:
1783:
1771:
1769:
1752:
1743:
1734:
1725:
1695:
1691:
1687:
1683:
1650:
1646:
1634:
1617:
1613:
1601:
1301:
1290:
1280:
1276:
1272:
1265:
1258:
1254:
1244:
1237:
1233:
1226:
1219:
1212:
1187:
1173:
1169:
1165:
1158:
1151:
1147:
1144:
1018:
1014:
1010:
999:
995:
993:must divide
984:
980:
970:
966:
962:
939:
935:
931:finite group
920:
916:
906:
890:
862:
618:
296:
289:
246:
223:
129:
61:
55:
50:
46:
42:
36:
2118:Mathematics
232:(stated by
2128:Categories
1909:PlanetMath
1830:, Oxford:
1766:References
1191:) and let
1008:such that
157:; that is
2071:Namesakes
1896:MathWorld
1661:φ
1534:≡
1520:φ
1444:φ
1429:∏
1414:φ
1379:φ
1364:∏
1360:≡
1336:φ
1321:∏
1095:≡
1039:φ
881:factoring
821:≡
778:φ
763:≡
707:φ
585:≡
579:≡
566:×
553:≡
540:×
514:≡
500:×
489:≡
440:≡
395:φ
224:In 1736,
187:≡
173:φ
141:φ
82:φ
2090:Category
1854:Springer
1788:Springer
1713:Archived
1572:See also
1271:, ... ,
1218:≡
1164:, ... ,
969:, ... ,
869:Internet
153:denotes
133:, where
974:modulo
950:coprime
803:, then
659:(where
639:modulo
344:, i.e.
67:coprime
2104:Portal
1860:
1838:
1812:
1794:
887:Proofs
234:Fermat
126:modulo
1638:See:
1605:See:
1594:Notes
1231:then
1225:(mod
1180:be a
956:then
944:. If
912:order
1858:ISBN
1836:ISBN
1810:ISBN
1792:ISBN
1770:The
1756:See
1300:mod
1289:mod
1252:and
1186:mod
679:and
267:and
65:are
59:and
1907:at
1545:mod
1473:mod
1257:= {
1150:= {
1119:mod
952:to
839:mod
774:mod
752:if
596:mod
484:222
451:mod
367:mod
357:222
330:222
198:mod
49:or
37:In
2130::
1893:.
1856:,
1834:,
1790:,
1698:).
1649:,
1616:,
1309::
1307:aR
1273:ax
1266:ax
1264:,
1259:ax
1255:aR
1236:=
1220:ax
1213:ax
1157:,
1013:=
1011:kM
965:,
925:.
748::
616:.
600:10
582:49
561:55
535:55
503:55
455:10
401:10
371:10
294:.
41:,
2106::
1937:e
1930:t
1923:v
1899:.
1705:.
1696:N
1692:N
1688:N
1684:N
1670:)
1667:n
1664:(
1651:8
1627:.
1618:8
1556:.
1552:)
1549:n
1542:(
1537:1
1529:)
1526:n
1523:(
1516:a
1500:i
1498:x
1484:,
1480:)
1477:n
1470:(
1463:i
1459:x
1453:)
1450:n
1447:(
1439:1
1436:=
1433:i
1423:)
1420:n
1417:(
1410:a
1406:=
1401:i
1397:x
1393:a
1388:)
1385:n
1382:(
1374:1
1371:=
1368:i
1355:i
1351:x
1345:)
1342:n
1339:(
1331:1
1328:=
1325:i
1302:n
1296:R
1291:n
1285:}
1283:)
1281:n
1279:(
1277:φ
1269:2
1262:1
1250:R
1245:n
1238:k
1234:j
1229:)
1227:n
1222:k
1215:j
1207:i
1205:x
1201:a
1197:n
1193:a
1188:n
1184:(
1178:}
1176:)
1174:n
1172:(
1170:φ
1166:x
1162:2
1159:x
1155:1
1152:x
1148:R
1130:.
1126:)
1123:n
1116:(
1111:1
1108:=
1103:M
1099:1
1090:M
1086:)
1080:k
1076:a
1072:(
1069:=
1064:M
1061:k
1057:a
1053:=
1048:)
1045:n
1042:(
1035:a
1021:)
1019:n
1017:(
1015:φ
1006:M
1002:)
1000:n
998:(
996:φ
991:k
987:)
985:n
981:a
976:n
971:a
967:a
963:a
958:a
954:n
946:a
942:)
940:n
938:(
936:φ
923:)
921:n
919:(
917:φ
907:n
901:n
897:n
873:n
859:.
846:)
843:n
836:(
829:y
825:a
816:x
812:a
790:)
787:)
784:n
781:(
771:(
766:y
760:x
736:a
716:)
713:n
710:(
687:n
667:a
647:n
627:a
603:)
593:(
588:9
574:2
570:7
557:1
548:2
544:7
531:)
525:4
521:7
517:(
509:2
506:+
497:4
493:7
480:7
458:)
448:(
443:1
435:4
431:7
410:4
407:=
404:)
398:(
374:)
364:(
353:7
326:7
305:n
275:n
255:a
242:n
238:n
209:.
205:)
202:n
195:(
190:1
182:)
179:n
176:(
169:a
130:n
113:1
91:)
88:n
85:(
78:a
62:a
56:n
34:.
20:)
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