# The Sum of Two Prime Numbers Greater than 2 is Even

In the previous posts, we have gone quite deep in delving and proving some complicated theorems. In this post, we go again the the basics. In this short proof, we show that the sum of two prime numbers, both greater than 2 is even.

Theorem

The sum of two primes, both greater than 2, is always even.

Proof

Let \$latex p\$ and \$latex q\$ be prime numbers both greater than 2. Then, both of them are odd numbers. This means that we can let \$latex p = 2r + 1\$ and \$latex q = 2s + 1\$ where \$latex r\$ and \$latex s\$ are positive integers.  Adding, we have

\$latex p + q = (2r + 1) + (2s + 1) = 2r + 2s + 2\$

Factoring out 2, we have

\$latex 2s + 2r + 2 = 2(s + r + 1)\$

Since \$latex 2(s + r + 1)\$ is divisible by \$latex 2\$, it is even. Therefore,

\$latex p + q = 2(s + r + 1)\$

is even.

Therefore, the sum of two primes both greater than 2 is even.

# An Intuitive Proof to The Infinitude of Primes

Introduction

Prime numbers are numbers with only two factors, \$latex 1\$ and itself. Eleven is prime since its factors are only \$latex 1\$ and \$latex 11\$. Numbers that are not prime are called composite. Fifteen is composite because it has more than \$latex 2\$ factors. The factors of \$latex 15\$ are \$latex 1\$, \$latex 3\$, \$latex 5\$, and \$latex 15\$.

White squares show primes from 1 to 100 (via mathandmultimedia.com).

As you count farther, you will observe that the number of primes are getting fewer and fewer. There are \$latex 168\$ primes between \$latex 1\$ and \$latex 1000\$, \$latex 135\$ primes between \$latex 1000\$ and \$latex 2000\$, \$latex 127\$ primes between \$latex 2000\$ and \$latex 3000\$, and \$latex 120\$ primes between \$latex 3000\$ and \$latex 4000\$. Now, the question is, “Is there a largest prime?” The equivalent to this question is, “Are the number of primes finite?” In this post, we are going to prove intuitively that the number of primes is infinitely many. Continue reading