Euler's theorem




This article is about Euler's theorem in number theory. For other uses, see List of topics named after Leonhard Euler.

In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that if n and a are coprime positive integers, then


aφ(n)≡1(modn){displaystyle a^{varphi (n)}equiv 1{pmod {n}}}a^{varphi (n)} equiv 1 pmod{n}

where φ(n){displaystyle varphi (n)}varphi (n) is Euler's totient function. (The notation is explained in the article modular arithmetic.) In 1736, Leonhard Euler published his proof of Fermat's little theorem,[1] 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.[2]


The converse of Euler's theorem is also true: if the above congruence is true, then a{displaystyle a}a and n{displaystyle n}n 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 n{displaystyle n}n. For example, consider finding the ones place decimal digit of 7222{displaystyle 7^{222}}{displaystyle 7^{222}}, i.e. 7222(mod10){displaystyle 7^{222}{pmod {10}}}{displaystyle 7^{222}{pmod {10}}}. Note that 7 and 10 are coprime, and φ(10)=4{displaystyle varphi (10)=4}{displaystyle varphi (10)=4}. So Euler's theorem yields 74≡1(mod10){displaystyle 7^{4}equiv 1{pmod {10}}}{displaystyle 7^{4}equiv 1{pmod {10}}}, and we get 7222≡74×55+2≡(74)55×72≡155×72≡49≡9(mod10){displaystyle 7^{222}equiv 7^{4times 55+2}equiv (7^{4})^{55}times 7^{2}equiv 1^{55}times 7^{2}equiv 49equiv 9{pmod {10}}}{displaystyle 7^{222}equiv 7^{4times 55+2}equiv (7^{4})^{55}times 7^{2}equiv 1^{55}times 7^{2}equiv 49equiv 9{pmod {10}}}.


In general, when reducing a power of a{displaystyle a}a modulo n{displaystyle n}n (where a{displaystyle a}a and n{displaystyle n}n are coprime), one needs to work modulo φ(n){displaystyle varphi (n)}varphi (n) in the exponent of a{displaystyle a}a:


if x≡y(modφ(n)){displaystyle xequiv y{pmod {varphi (n)}}}x equiv y pmod{varphi(n)}, then ax≡ay(modn){displaystyle a^{x}equiv a^{y}{pmod {n}}}a^x equiv a^y pmod{n}.

Euler's theorem is sometimes cited as forming the basis of the RSA encryption system, however it is insufficient (and unnecessary) 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 kφ(n)+1{displaystyle kvarphi (n)+1}{displaystyle kvarphi (n)+1}, for some positive integer k{displaystyle k}k. In the case that the original number is relatively prime to n{displaystyle n}n, Euler's theorem then guarantees that the decrypted output number is equal to the original input number, giving back the plaintext. However, because n{displaystyle n}n is a product of two distinct primes, p{displaystyle p}p and q{displaystyle q}q, when the number encrypted is a multiple of p{displaystyle p}p or q{displaystyle q}q, 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 n{displaystyle n}n, and so Euler's theorem is neither sufficient nor necessary.




Contents






  • 1 Proofs


  • 2 Euler quotient


  • 3 See also


  • 4 Notes


  • 5 References


  • 6 External links





Proofs


1. Euler's theorem can be proven using concepts from the theory of groups:[3]
The residue classes (mod n) that are coprime to n form a group under multiplication (see the article Multiplicative group of integers modulo n for details). The order of that group is φ(n){displaystyle varphi (n)}varphi (n) where φ{displaystyle varphi }varphi 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 φ(n){displaystyle varphi (n)}varphi (n). If a{displaystyle a}a is any number coprime to n{displaystyle n}n then a{displaystyle a}a is in one of these residue classes, and its powers a,a2,…,ak≡1(modn){displaystyle a,a^{2},ldots ,a^{k}equiv 1{pmod {n}}}{displaystyle a,a^{2},ldots ,a^{k}equiv 1{pmod {n}}} are a subgroup. Lagrange's theorem says k{displaystyle k}k must divide φ(n){displaystyle varphi (n)}varphi (n), i.e. there is an integer M{displaystyle M}M such that kM=φ(n){displaystyle kM=varphi (n)}{displaystyle kM=varphi (n)}. But then,


aφ(n)=akM=(ak)M≡1M=1≡1(modn).{displaystyle a^{varphi (n)}=a^{kM}=(a^{k})^{M}equiv 1^{M}=1equiv 1{pmod {n}}.}{displaystyle a^{varphi (n)}=a^{kM}=(a^{k})^{M}equiv 1^{M}=1equiv 1{pmod {n}}.}

2. There is also a direct proof:[4][5] Let R={x1,x2,…,xφ(n)}{displaystyle R=lbrace x_{1},x_{2},ldots ,x_{varphi (n)}rbrace }{displaystyle R=lbrace x_{1},x_{2},ldots ,x_{varphi (n)}rbrace } be a reduced residue system (mod n{displaystyle n}n) and let a{displaystyle a}a be any integer coprime to n{displaystyle n}n. The proof hinges on the fundamental fact that multiplication by a{displaystyle a}a permutes the xi{displaystyle x_{i}}x_{i}: in other words if axj≡axk(modn){displaystyle ax_{j}equiv ax_{k}{pmod {n}}}{displaystyle ax_{j}equiv ax_{k}{pmod {n}}} then j=k{displaystyle j=k}j = k. (This law of cancellation is proved in the article multiplicative group of integers modulo n.[6]) That is, the sets R{displaystyle R}R and aR={ax1,ax2,…,axφ(n)}{displaystyle aR=lbrace ax_{1},ax_{2},ldots ,ax_{varphi (n)}rbrace }{displaystyle aR=lbrace ax_{1},ax_{2},ldots ,ax_{varphi (n)}rbrace }, considered as sets of congruence classes (mod n{displaystyle n}n), are identical (as sets - they may be listed in different orders), so the product of all the numbers in R{displaystyle R}R is congruent (mod n{displaystyle n}n) to the product of all the numbers in aR{displaystyle aR}{displaystyle aR}:



∏i=1φ(n)xi≡∏i=1φ(n)axi=aφ(n)∏i=1φ(n)xi(modn),{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}},}{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}},} and using the cancellation law to cancel the xi{displaystyle x_{i}}x_{i}s gives Euler's theorem:

aφ(n)≡1(modn).{displaystyle a^{varphi (n)}equiv 1{pmod {n}}.}<br /> a^{varphi(n)}equiv 1 pmod{n}.<br />


Euler quotient


The Euler quotient of an integer a with respect to n is defined as:


qn(a)=aφ(n)−1n{displaystyle q_{n}(a)={frac {a^{varphi (n)}-1}{n}}}{displaystyle q_{n}(a)={frac {a^{varphi (n)}-1}{n}}}

The special case of Euler quotient is Fermat quotient, it happens when n is prime.


A number n coprime to a which divides qn(a){displaystyle q_{n}(a)}{displaystyle q_{n}(a)} is called generalized Wieferich number to base a. In a special case, an odd number n which divides qn(2){displaystyle q_{n}(2)}{displaystyle q_{n}(2)} is called Wieferich number.































































































































































a
 numbers n coprime to a which divides qn(a){displaystyle q_{n}(a)}{displaystyle q_{n}(a)} (searched up to 1048576)
OEIS sequence
1
 1, 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, ... (all natural numbers)
A000027
2
 1, 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, ...
A077816
3
 1, 11, 22, 44, 55, 110, 220, 440, 880, 1006003, 2012006, 4024012, 11066033, 22132066, 44264132, 55330165, 88528264, 110660330, 221320660, 442641320, 885282640, 1770565280, 56224501667, 112449003334, ...
A242958
4
 1, 1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, 127881, 136929, 157995, 228215, 298389, 410787, 473985, 684645, 895167, ...
5
 1, 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, ...
A242959
6
 1, 66161, 330805, 534851, 2674255, 3152573, 10162169, 13371275, 50810845, 54715147, 129255493, 148170931, 254054225, 273575735, 301121113, 383006029, 646277465, ...
A241978
7
 1, 4, 5, 10, 20, 40, 80, 491531, 983062, 1966124, 2457655, 3932248, 4915310, 6389903, 9339089, 9830620, 12288275, 12779806, 18678178, 19169709, 19661240, 24576550, 25559612, ...
A242960
8
 1, 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, ...
9
 1, 2, 4, 11, 22, 44, 55, 88, 110, 220, 440, 880, 1760, 1006003, ...
10
 1, 3, 487, 1461, 4383, 13149, 39447, 118341, 355023, 56598313, 169794939, 509384817, ...
A241977
11
 1, 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, ...
A253016
12
 1, 2693, 123653, 1812389, 2349407, 12686723, 201183431, 332997529, ...
A245529
13
 1, 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, ...
A257660
14
 1, 29, 353, 3883, 10237, 19415, 112607, 563035, ...
15
 1, 4, 8, 29131, 58262, 116524, 233048, 466096, ...
16
 1, 1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, 127881, 136929, 157995, 228215, 298389, 410787, 473985, 684645, 895167, ...
17
 1, 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, ...
18
 1, 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, ...
19
 1, 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, ...
20
 1, 281, 1967, 5901, 46457, ...
21
 1, 2, ...
22
 1, 13, 39, 673, 2019, 4711, 8749, 14133, 26247, 42399, 61243, 78741, 183729, 551187, ...
23
 1, 4, 13, 26, 39, 52, 78, 104, 156, 208, 312, 624, 1248, ...
24
 1, 5, 25633, 128165, ...
25
 1, 2, 4, 20771, 40487, 41542, 80974, 83084, 161948, 166168, 323896, 643901, ...
26
 1, 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, ...
27
 1, 11, 22, 44, 55, 110, 220, 440, 880, 1006003, ...
28
 1, 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, ...
29
 1, 2, ...
30
 1, 7, 160541, ...

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


2, 5, 8, 7, 7, 17, 18, 15, 26, 7, 3, 17, 19, 19, 26, 31, 38, 53, 28, 7, 19, 3, 28, 17, 57, 19, 80, 19, 14, 107, 115, 63, 118, 65, 18, 53, 18, 69, 19, 7, 51, 19, 19, 3, 26, 63, 53, 17, 18, 57, ... (sequence A250206 in the OEIS)


See also



  • Carmichael function

  • Euler's criterion

  • Fermat's little theorem

  • Wilson's theorem



Notes





  1. ^ See:

    • Leonhard Euler (presented: August 2, 1736; published: 1741) "Theorematum quorundam ad numeros primos spectantium demonstratio" (A proof of certain theorems regarding prime numbers), Commentarii academiae scientiarum Petropolitanae, 8 : 141–146.

    • For further details on this paper, including an English translation, see: The Euler Archive.




  2. ^ See:

    • L. Euler (published: 1763) "Theoremata arithmetica nova methodo demonstrata" (Proof of a new method in the theory of arithmetic), Novi Commentarii academiae scientiarum Petropolitanae, 8 : 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, φ(n){displaystyle varphi (n)}varphi (n), is not named but referred to as "numerus partium ad N primarum" (the number of parts prime to N; that is, the number of natural numbers that are smaller than N and relatively prime to N).

    • For further details on this paper, see: The Euler Archive.

    • For a review of Euler's work over the years leading to Euler's theorem, see: Ed Sandifer (2005) "Euler's proof of Fermat's little theorem"




  3. ^ Ireland & Rosen, corr. 1 to prop 3.3.2


  4. ^ Hardy & Wright, thm. 72


  5. ^ Landau, thm. 75


  6. ^ See Bézout's lemma




References


The Disquisitiones Arithmeticae 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.




  • Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English) (1986), Disquisitiones Arithemeticae (Second, corrected edition), New York: Springer, ISBN 0-387-96254-9.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  • Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8


  • Hardy, G. H.; Wright, E. M. (1980), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford University Press, ISBN 978-0-19-853171-5


  • Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, ISBN 0-387-97329-X


  • Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea



External links



  • Weisstein, Eric W. "Euler's Totient Theorem". MathWorld.


  • Euler-Fermat Theorem at PlanetMath









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