Euler's theorem

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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 a raised to the power of the totient of n is congruent to one, modulo n, or:

where is Euler's totient function. 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 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 (where and are coprime), one needs to work modulo in the exponent of :

if , then .

Euler's theorem underlies RSA cryptosystem, which is widely used in Internet communications. In this cryptosystem, Euler's theorem is used with n being a product of two large prime numbers, and the security of the system is based on the difficulty of factoring such an integer.

Proofs[edit]

1. Euler's theorem can be proven using concepts from the theory of groups:[3] The residue classes modulo 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). 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). If a is any number coprime to n then a is in one of these residue classes, and its powers a, a2, ... , ak modulo n form a subgroup of the group of residue classes, with ak ≡ 1 (mod n). Lagrange's theorem says k must divide φ(n), i.e. there is an integer M such that kM = φ(n). This then implies,

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

and using the cancellation law to cancel each xi gives Euler's theorem:

See also[edit]

Notes[edit]

  1. ^ See:
  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, , 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[edit]

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. (translated into English) (1986), Disquisitiones Arithemeticae (Second, corrected edition), New York: Springer, ISBN 0-387-96254-9
  • Gauss, Carl Friedrich; Maser, H. (translated 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[edit]