User:Kip/Sandbox

Theorem
Let $m\in\Z_{>1}$ be a positive integer greater than one.

Let $x\in\Z_{>1}$ be a positive integer greater than one.

Let $A\in\Z_{>0}$ be a positive integer coprime with $m$.

Then:
 * $\displaystyle A^{x}\equiv a \pmod m$

Is an $n^{th}$ root of unity modulo $m$ where:
 * $\displaystyle n=\frac{\phi(m)}{gcd(\phi(m),x)}$

$\phi(m)$ is Euler's Totient function of the modulus and $gcd(\phi(m),x)$ is the greatest common divisor of the totient and the power.

Proof
Let $\alpha=\frac{x}{gcd(\phi(m),x)}\in\Z_{>0}$ be a positive integer.
 * $\displaystyle A^{\alpha \phi(m)}\equiv a^n\equiv 1\pmod m$

Square Root of Unity Addition Theorem
Let $p\in\Z_{>0}$ be a prime number greater. Let $x,y,z\in\Z_{>1}$ be positive integers greater than one for which:
 * $2gcd(p-1,x)=2gcd(p-1,y)=n gcd(p-1,z)=p-1$

Then
 * $A^x+B^y\ne C^z \pmod m$

Proof

 * $A^{2x}+2A^xB^y+B^{2y}\equiv C^{2z} \pmod m$

Since the binomial sum is:
 * $\displaystyle \sum_{k\mathop =0}^n \binom n k =2^n$
 * $2A^xB^y\equiv -1 \pmod m$
 * $1\pm 2\equiv 0 \pmod m$
 * $mk=m-1$

Or
 * $m=3$

Theorem
Let $X$ be a continuous random variable with the exponential distribution with parameter $\beta$.

Then the expectation of $X$ is given by:
 * $E \left({X}\right) = \beta$

Proof
The expectation is
 * $\displaystyle E \left({X}\right) := \int_{x \mathop \in \Omega_X} x \ f_X \left({x}\right) \ \mathrm d x$

Which, for the Exponential Distribution is
 * $\displaystyle E \left({X}\right) = \int_{0}^{\infty} x \frac{1}{\beta} e^{-\frac{x}{\beta}} \ \mathrm d x$

Substituting $u=\frac{x}{\beta}$:
 * $\displaystyle E \left({X}\right) = \beta\int_{0}^{\infty} u e^{-u} \ \mathrm d u$

The integral evaluates to
 * $\displaystyle E \left({X}\right) = -\beta (u+1) e^{-u}|_0^{\infty} $

Using L'Hôpital's Rule, the limit ends up as
 * $E \left({X}\right) = \beta$