Euler's theorem

Probably the most stupid name you could give a theorem. Leonhard Euler was a an extremely productive mathematician who made contributions to practically all fields of mathematics. The result is that there is not just one Euler's theorem, but lots and lots. After a while people started naming theorems after the next guy to investigate it after Euler had discovered it.

Some examples:

Euler's theorem (number theory):
If a and n are coprime then a^φ(n) ≡ 1 (mod n). (Discussed in detail below by Noether.)

Euler's theorem (vector calculus):
f : Rⁿ → R is homogeneous function of degree n iff (x.∇)f = nf.

Euler's theorem (kinematics):
For any rigid body rotation there is an instantaneous axis of rotation (ie. a straight line of points instantenously at rest).

The long list of mathematical concepts named after Euler also includes items such as the Euler characteristic, the Euler formula, the Euler line, Euler's constant, the Euler momentum equation and the Euler-Lagrange equations.

As krimson has mentioned, calling a result Euler's theorem somewhat under-specifies it. However, here is a theorem that usually goes by that name. It's worth noting that although this result is simple it is used in an important way in the RSA cryptosystem. We need a definition.

Definition If a,b,n are integers with n>0 we say that a is congruent to b modulo n if n divides a-b.

So for example 4 is congruent to 1 modulo 3. We write phi for the Euler Phi function.

Euler's Theorem If a and n are coprime positive integers then

a^phi(n) is congruent to 1 modulo n.

Before we prove this result, note that Fermat's Little Theorem is the special case of this result when n is prime.

Proof: We'll use a little bit of technology to prove the theorem. Consider Z_n the ring of integers modulo n. Have a look at that writeup because we'll use some results and notation from it. It is shown there that for an integer a we have a is a unit in Z_n if and only if (a,n)=1. So we see from the definition of the Euler Phi function that the multiplicative group of units in the ring Z_n has order phi(n). By Lagrange's theorem if we raise an element in a finite group to the power the group order we get the identity element. Thus, if a is as in the statement it satisfies

a^phi(n) = 1

The result follows since for two integers x and y we have x=y if and only if x is congruent to y modulo n.