# The even primes

A natural number *k* is said to *prime* if it is divisible only by 1 and itself. Prime numbers have become very important to modern society, and encryption in particular is based on sophisticated principles of the prime numbers.

This page contains a number of theorems and lemmas on a subset of the prime numbers, namely the set of *even* prime numbers. No proofs are given, as I wish the reader to experience the joy of reproducing these fundamental truths himself.

Do you know any other theorems about the even primes? Feel free to send them to me, and I’ll put them up here.

## Finiteness

The set of even primes is quite finite.

## Countability

The set of even primes is countable. In fact, the even primes are *very easily* countable.

## Henriksens’ Identity

The sum of any non-empty set of even primes is equal to one plus the number of elements in the set.

## Convergence

When summing a sequence of *n* even primes, as *n* goes towards infinity, the sum of the sequence will (very rapidly) approach *2n*.

## Dahlgaard’s Theorem

The equation *a^p + b^p = c^p* has integer solutions for all even primes *p*.

## The Malmkjær-Mørch Conjecture

The product of all even even prime numbers is equal to the sum of all even prime numbers.

## Lund’s Last Theorem

Given an even prime *p*, and a superperfect number *s*, then *p* will be a proper divisor of *s*.

## Serup’s Odd Principle

No odd integer greater than 1 can be expressed as the sum of two even primes.

## Knudsen’s Generating Lemma

Given any even prime number *p*, and any uneven prime number *q*, *p* modulo *q* will result in another even prime number.

## Closure under factorial (originally shown by Martin Haugaard)

For any even prime *p*, *p!* is also an even prime.

## Jensen’s Identity

The sum of any nonempty subset of the even prime numbers, divided by the number of elements in the subset, equals the smallest element in the subset.

## The Torben Mogensen Properties

No triangle where the sides are three even primes is a Pythagorean triangle.

For any three different even primes, *p*, *q* and *r*, it holds that *p^3 + q^3 = r^3*.

## Strake’s Generator

Every number *(2^n)-1*, where *n* is an even prime, is itself prime.

## The Strake-Henriksen Generator

Every number *2^(n-1)*, where *n* is an even prime, is itself prime.

*Note:* The discovery of this groundbreaking result was the result of an erroneous placement of parentheses by the editor during the first publication of *Strake’s Generator*.