Euler's theorem


In number theory, Euler's theorem states that if n and a are coprime positive integers, then
where is Euler's totient function. In 1736, Leonhard Euler published his proof of Fermat's little theorem, which Fermat had presented without proof. Subsequently, Euler presented other proofs of the theorem, culminating with "Euler's theorem" in his paper of 1763, in which he attempted to find the smallest exponent for which Fermat's little theorem was always true.
The converse of Euler's theorem is also true: if the above congruence is true, then and must be coprime.
The theorem is a generalization of Fermat's little theorem, and is further generalized by Carmichael's theorem.
The theorem may be used to easily reduce large powers modulo. For example, consider finding the ones place decimal digit of, i.e.. The integers 7 and 10 are coprime, and. So Euler's theorem yields, and we get.
In general, when reducing a power of modulo , one needs to work modulo in the exponent of :
Euler's theorem is sometimes cited as forming the basis of the RSA encryption system, however it is insufficient to use Euler's theorem to certify the validity of RSA encryption. In RSA, the net result of first encrypting a plaintext message, then later decrypting it, amounts to exponentiating a large input number by, for some positive integer. In the case that the original number is relatively prime to, Euler's theorem then guarantees that the decrypted output number is equal to the original input number, giving back the plaintext. However, because is a product of two distinct primes, and, when the number encrypted is a multiple of or, Euler's theorem does not apply and it is necessary to use the uniqueness provision of the Chinese Remainder Theorem. The Chinese Remainder Theorem also suffices in the case where the number is relatively prime to, and so Euler's theorem is neither sufficient nor necessary.

Proofs

1. Euler's theorem can be proven using concepts from the theory of groups:
The residue classes modulo that are coprime to form a group under multiplication. The order of that group is where denotes Euler's totient function. Lagrange's theorem states that the order of any subgroup of a finite group divides the order of the entire group, in this case. If is any number coprime to then is in one of these residue classes, and its powers modulo form a subgroup of the group of residue classes, with. Lagrange's theorem says must divide, i.e. there is an integer such that. This then implies,
2. There is also a direct proof: Let be a reduced residue system and let be any integer coprime to. The proof hinges on the fundamental fact that multiplication by permutes the : in other words if then. That is, the sets and, considered as sets of congruence classes, are identical, so the product of all the numbers in is congruent to the product of all the numbers in :

Euler quotient

The Euler quotient of an integer a with respect to n is defined as:
The special case of an Euler quotient when n is prime is called a Fermat quotient.
Any odd number n that divides is called a Wieferich number. This is equivalent to saying that 2 ≡ 1. As a generalization, any number n that is coprime to a positive integer a, and such that n divides, is called a Wieferich number to base a. This is equivalent to saying that a ≡ 1.
anumbers n coprime to a that divide OEIS sequence
11, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,...
21, 1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, 127881, 136929, 157995, 228215, 298389, 410787, 473985, 684645, 895167, 1232361, 2053935, 2685501, 3697083, 3837523, 6161805, 11512569,...
31, 11, 22, 44, 55, 110, 220, 440, 880, 1006003, 2012006, 4024012, 11066033, 22132066, 44264132, 55330165, 88528264, 110660330, 221320660, 442641320, 885282640, 1770565280, 56224501667, 112449003334,...
41, 1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, 127881, 136929, 157995, 228215, 298389, 410787, 473985, 684645, 895167,...
51, 2, 20771, 40487, 41542, 80974, 83084, 161948, 643901, 1255097, 1287802, 1391657, 1931703, 2510194, 2575604, 2783314, 3765291, 3863406, 4174971, 5020388, 5151208, 5566628, 7530582, 7726812, 8349942, 10040776, 11133256, 15061164, 15308227, 15453624, 16699884,...
61, 66161, 330805, 534851, 2674255, 3152573, 10162169, 13371275, 50810845, 54715147, 129255493, 148170931, 254054225, 273575735, 301121113, 383006029, 646277465,...
71, 4, 5, 10, 20, 40, 80, 491531, 983062, 1966124, 2457655, 3932248, 4915310, 6389903, 9339089, 9830620, 12288275, 12779806, 18678178, 19169709, 19661240, 24576550, 25559612,...
81, 3, 1093, 3279, 3511, 7651, 9837, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, 127881, 136929, 157995, 206577, 228215, 284391, 298389, 383643, 410787, 473985, 684645, 895167,...
91, 2, 4, 11, 22, 44, 55, 88, 110, 220, 440, 880, 1760, 1006003,...
101, 3, 487, 1461, 4383, 13149, 39447, 118341, 355023, 56598313, 169794939, 509384817,...
111, 71, 142, 284, 355, 497, 710, 994, 1420, 1491, 1988, 2485, 2840, 2982, 3976, 4970, 5680, 5964, 7455, 9940, 11928, 14910, 19880, 23856, 29820, 39760, 59640, 79520, 119280, 238560, 477120,...
121, 2693, 123653, 1812389, 2349407, 12686723, 201183431, 332997529,...
131, 2, 863, 1726, 3452, 371953, 743906, 1487812, 1747591, 1859765, 2975624, 3495182, 3719530, 5242773, 6990364, 7439060, 8737955, 10485546, 14878120, 15993979, 17475910, 20971092, 26213865, 29756240, 31987958, 34951820, 41942184, 47981937, 52427730, 59512480,...
141, 29, 353, 3883, 10237, 19415, 112607, 563035,...
151, 4, 8, 29131, 58262, 116524, 233048, 466096,...
161, 1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, 127881, 136929, 157995, 228215, 298389, 410787, 473985, 684645, 895167,...
171, 2, 3, 4, 6, 8, 12, 24, 48, 46021, 48947, 92042, 97894, 138063, 146841, 184084, 195788, 230105, 276126, 293682, 368168, 391576, 414189, 460210, 552252, 587364, 598273, 690315, 736336, 783152, 828378, 920420,...
181, 5, 7, 35, 37, 49, 185, 245, 259, 331, 1295, 1655, 1813, 2317, 3641, 8275, 9065, 11585, 12247, 16219, 18205, 25487, 33923, 57925, 61235, 81095, 85729, 91025, 127435, 134717, 169615, 178409, 237461, 306175, 405475, 428645, 455125, 600103, 637175, 673585, 892045, 943019,...
191, 3, 6, 7, 12, 13, 14, 21, 26, 28, 39, 42, 43, 49, 52, 63, 78, 84, 86, 91, 98, 104, 117, 126, 129, 137, 147, 156, 168, 172, 182, 196, 234, 252, 258, 273, 274, 294, 301, 312, 364, 387, 411, 441, 468, 504, 516, 546, 548, 559, 588, 602, 624, 637, 728, 774, 819, 822, 882, 903, 936, 959, 1032, 1092, 1096, 1118, 1176, 1204, 1274, 1456, 1548, 1638, 1644, 1677, 1764, 1781, 1806, 1872, 1911, 1918, 2107, 2184, 2192, 2236, 2329, 2408, 2457, 2548, 2709, 2877, 3096, 3276, 3288, 3354, 3528, 3562, 3612, 3822, 3836, 3913, 4214, 4368, 4472, 4658, 4914, 5031, 5096, 5343, 5418, 5733, 5754, 5891, 6321, 6552, 6576, 6708, 6713, 6987, 7124, 7224, 7644, 7672, 7826, 8127, 8428, 8631, 8736, 8944, 9316, 9828, 10062, 10192, 10686, 10836, 11466, 11508, 11739, 11782, 12467, 12642, 13104, 13152, 13416, 13426, 13974, 14248, 14448, 14749, 15093, 15288, 15344, 15652, 16029, 16254, 16303, 16856, 17199, 17262, 17673, 18632, 18963, 19656, 20124, 20139, 21372, 21672, 22932, 23016, 23478, 23564, 24934, 25284, 26208, 26832, 26852, 27391, 27948, 28496, 29498, 30186, 30277, 30576, 30688, 31304, 32058, 32508, 32606, 34398, 34524, 35217, 35346, 37264, 37401, 37926, 39312, 40248, 40278, 41237, 42744, 43344, 44247, 45864, 46032, 46956, 47128, 48909, 49868, 50568, 53019, 53664, 53704, 54782, 55896, 56889, 56992, 58996, 60372, 60417, 60554, 61152, 62608, 64116, 65016, 65212, 68796, 69048, 70434, 70692, 74528, 74802, 75852, 76583, 78624, 80496, 80556, 82173, 82474, 85488, 87269, 88494, 90831, 91728, 92064, 93912, 94256, 97818, 99736, 100147, 101136, 105651, 106038, 107408, 109564, 111792, 112203, 113778, 113984, 114121, 117992, 120744, 120834, 121108, 123711, 125216, 128232, 130032, 130424, 132741, 137592, 138096, 140868, 141384, 146727, 149056, 149604, 151704, 153166, 160992, 161112, 164346, 164948, 170976, 174538, 176988, 181662, 183456, 184128, 187824, 188512, 191737, 195636, 199472, 200294, 211302, 211939, 212076, 214816, 219128, 223584, 224406, 227556, 228242, 229749, 241488, 241668, 242216, 246519, 247422, 256464, 260848, 261807, 265482, 272493, 275184, 276192, 281736, 282768, 288659, 293454, 298112, 299208, 300441, 303408, 306332, 316953, 322224, 328692, 329896, 336609, 341952, 342363, 349076, 353976, 363324, 371133, 375648, 383474, 391272, 398223, 398944, 400588, 422604, 423878, 424152, 438256, 447168, 448812, 455112, 456484, 459498, 482976, 483336, 484432, 493038, 494844, 512928, 521696, 523614, 530964, 536081, 544986, 550368, 552384, 563472, 565536, 575211, 577318, 586908, 596224, 598416, 600882, 612664, 633906, 635817, 644448, 657384, 659792, 673218, 683904, 684726, 689247, 698152, 701029, 707952, 726648, 739557, 742266, 751296, 766948, 782544, 785421, 796446, 797888, 801176, 845208, 847756, 848304, 865977, 876512, 894336, 897624, 901323, 910224, 912968, 918996, 966672, 968864, 986076, 989688, 1025856, 1027089, 1043392, 1047228,...
201, 281, 1967, 5901, 46457,...
211, 2,...
221, 13, 39, 673, 2019, 4711, 8749, 14133, 26247, 42399, 61243, 78741, 183729, 551187,...
231, 4, 13, 26, 39, 52, 78, 104, 156, 208, 312, 624, 1248,...
241, 5, 25633, 128165,...
251, 2, 4, 20771, 40487, 41542, 80974, 83084, 161948, 166168, 323896, 643901,...
261, 3, 5, 9, 15, 45, 71, 213, 355, 497, 639, 1065, 1491, 1775, 2485, 3195, 4473, 5325, 7455, 12425, 13419, 15975, 22365, 37275, 67095, 111825, 335475,...
271, 11, 22, 44, 55, 110, 220, 440, 880, 1006003,...
281, 3, 9, 19, 23, 57, 69, 171, 207, 253, 437, 513, 759, 1265, 1311, 1539, 2277, 3795, 3933, 4807, 11385, 11799, 14421, 24035, 35397, 43263, 72105, 129789, 216315, 389367, 648945,...
291, 2,...
301, 7, 160541,...

The least base b > 1 which n is a Wieferich number are