Euler's theorem

Template:Short description 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 $a^{\phi (n)}$ is congruent to $1$ modulo n, where $\phi$ denotes Euler's totient function; that is

$a^{\phi (n)}\equiv 1(\mathrm{mod}n).$

In 1736, Leonhard Euler published a proof of Fermat's little theorem (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where Template:Mvar 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 Template:Mvar is not prime.

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. $7^{222}(\mathrm{mod}10)$. The integers 7 and 10 are coprime, and $\phi (10)=4$. So Euler's theorem yields $7^4\equiv 1(\mathrm{mod}10)$, and we get $7^{222}\equiv 7^{4\times 55+2}\equiv (7^4{)}^{55}\times 7^2\equiv 1^{55}\times 7^2\equiv 49\equiv 9(\mathrm{mod}10)$.

In general, when reducing a power of $a$ modulo $n$ (where $a$ and $n$ are coprime), one needs to work modulo $\phi (n)$ in the exponent of $a$:

if $x\equiv y(\mathrm{mod}\phi (n))$, then $a^x\equiv a^y(\mathrm{mod}n)$.

Euler's theorem underlies the RSA cryptosystem, which is widely used in Internet communications. In this cryptosystem, Euler's theorem is used with Template:Mvar 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

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

$a^{\phi (n)}=a^{kM}=(a^k{)}^M\equiv 1^M=1(\mathrm{mod}n).$

2. There is also a direct proof: Let R = {x₁, x₂, ... , x_φ(n)} be a reduced residue system (mod n) and let Template:Mvar be any integer coprime to Template:Mvar. The proof hinges on the fundamental fact that multiplication by Template:Mvar permutes the Template:Mvar: in other words if ax_j ≡ ax_k (mod n) then j = k. (This law of cancellation is proved in the article Multiplicative group of integers modulo n.) That is, the sets Template:Mvar and aR = {ax₁, ax₂, ... , ax_φ(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 Template:Mvar is congruent (mod n) to the product of all the numbers in Template:Mvar:

$\prod _{i=1}^{\phi (n)}{x}_i\equiv \prod _{i=1}^{\phi (n)}a{x}_i=a^{\phi (n)}\prod _{i=1}^{\phi (n)}{x}_i(\mathrm{mod}n),$ and using the cancellation law to cancel each Template:Mvar gives Euler's theorem:

$a^{\phi (n)}\equiv 1(\mathrm{mod}n).$

See also

• Carmichael number
• Euler's criterion
• Wilson's theorem

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.

External links

• Euler-Fermat Theorem at PlanetMath