Euler's Theorem

In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, and $\varphi(n)$ is Euler's totient function, then raised to the power $\varphi(n)$ is congruent to modulo ; that is
:$a^ \equiv 1 \pmod.$

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 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 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 Carmichael's theorem.

The theorem may be used to easily reduce large powers modulo $n$. For example, consider finding the ones place decimal digit of $7^$, i.e. $7^ \pmod$. The integers 7 and 10 are coprime, and $\varphi(10) = 4$. So Euler's theorem yields $7^4 \equiv 1 \pmod$, and we get $7^ \equiv 7^ \equiv (7^4)^ \times 7^2 \equiv 1^ \times 7^2 \equiv 49 \equiv 9 \pmod$.

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 $x \equiv y \pmod$, then $a^x \equiv a^y \pmod$.

Euler's theorem underlies the RSA cryptosystem, which is widely used in Internet communications. In this cryptosystem, Euler's theorem is used with 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 that are coprime to form a group under multiplication (see the article Multiplicative group of integers modulo ''n'' for details). The order of that group is . 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,
:$a^ = a^ = (a^)^M \equiv 1^M =1 \equiv 1 \pmod.$

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 . (This law of cancellation is proved in the article Multiplicative group of integers modulo ''n''.See Bézout's lemma) That is, the sets and , considered as sets of congruence classes (), are identical (as sets—they may be listed in different orders), so the product of all the numbers in is congruent () to the product of all the numbers in :
:$\prod_^ x_i \equiv \prod_^ ax_i = a^\prod_^ x_i \pmod,$ and using the cancellation law to cancel each gives Euler's theorem:

:$a^\equiv 1 \pmod.$

See also

* Carmichael function
* Euler's criterion
* Fermat's little theorem
* Wilson's theorem

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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External links

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