---
title: The even primes
---

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](http://en.wikipedia.org/wiki/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_.