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Euler's theorem

Theorem on modular exponentiation

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^{\varphi(n)} is congruent to 1 modulo n, where \varphi denotes Euler's totient function; that is :a^{\varphi (n)} \equiv 1 \pmod{n}.

In 1736, Leonhard Euler published a proof of Fermat's little theoremSee:

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. (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n 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 n is not prime.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, \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"

The converse of Euler's theorem is also true: if the above congruence is true, then a and n must be coprime.

The theorem is further generalized by some of Carmichael's theorems.

The theorem may be used to easily reduce large powers modulo n. For example, consider finding the ones place decimal digit of 7^{222}, i.e. 7^{222} \pmod{10}. The integers 7 and 10 are coprime, and \varphi(10) = 4. So Euler's theorem yields 7^4 \equiv 1 \pmod{10}, and we get 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}.

In general, when reducing a power of a modulo n (where a and n are coprime), one needs to work modulo \varphi(n) in the exponent of a: :if x \equiv y \pmod{\varphi(n)}, then a^x \equiv a^y \pmod{n}.

Euler's theorem underlies the 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

Euler's theorem can be proven using concepts from the theory of groups: 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, a, ... , a modulo n form a subgroup of the group of residue classes, with a ≡ 1 (mod n). Lagrange's theorem says k must divide φ(n), i.e. there is an integer M such that . This then implies, :a^{\varphi(n)} = a^{kM} = (a^{k})^M \equiv 1^M =1 \pmod{n}.
There is also a direct proof: Let 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 x: in other words if ax ≡ ax (mod n) then . (This law of cancellation is proved in the article Multiplicative group of integers modulo n.) That is, the sets R and , 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: : \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 each x gives Euler's theorem:

: a^{\varphi(n)}\equiv 1 \pmod{n}.

Notes

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.

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References

Ireland & Rosen, corr. 1 to prop 3.3.2
Hardy & Wright, thm. 72
Landau, thm. 75
See [[Bézout's lemma]]

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