Two types of odd primes:4k+1 and 4k+3

Welcome to Math Year 2014. Like with Math Year 2013, I will be posting here from time to time. I will be doing election prediction work on the Senate and Governor's races when they come around and may do another series on climate change as well. Math Year 2013 will be available as a link at the top of the page. That blog started with a question about whether 2013 was a prime or not. (The answer is no, 2013 = 3 × 11 × 61.) Today, we will look a way to split the odd primes into two groups, the 4k+1 primes and the 4k+3 primes.

One of the "easiest" facts about the primes is that 2 is the only even prime. Any even number bigger than 2 is divisible by 2, so it isn't prime. Let's look at the odd numbers less than 30 to see which ones are prime.

3 is prime.
5 is prime.
7 is prime.
9 = 3 × 3, so no.
11 is prime.
13 is prime.
15 = 3 × 5, so no.
17 is prime.
19 is prime.
21 = 3 × 7, so no.
23 is prime.
25 = 5 × 5, so no
27 = 3 × 3 × 3, so no.
29 is prime.

If we divide any odd number by 4, the remainder will have to be odd, either 1 or 3. Looking at the primes on this short list, here are the 4k+1 primes.

4k+1 primes less than 30: 5, 13, 17, 29

And these are the 4k+3 primes less than 30: 3, 7, 11, 19, 23

Here is an unusual fact about the 4k+1 primes. Every one is the sum of two perfect squares. It is not easy to prove this, but it's not hard to illustrate with the early examples.

5 = 1² + 2²
13 = 2² + 3² 
17 = 1² + 4²
29 = 2² + 5²

In all cases, we will have prime = even² + odd², since even² + even² = even and odd² + odd² = even, and all the primes we are looking at must be odd, since they are all larger than 2.

Let's look at the primes between 2000 and 2100, numbers we found last year on January 2. We will split them into 4k+1 and 4k+3 primes.

4k+ 1 primes: 2017, 2029, 2053, 2069, 2081, 2089
4k+3 primes: 2003, 2011, 2027, 2039, 2063, 2083, 2087, 2099

It's not as easy to find the two squares that add up to these larger numbers, but here they are.
2017 = 9² + 44² = 81 + 1936
2029 = 2² + 45² = 4 + 2025
2053 = 17² + 42² = 289 + 1764
2069 = 25² + 38² = 625 + 1444
2081 = 20² + 41² = 400 + 1681
2089 = 8² + 45² = 64 + 2025

Another strange fact about 4k+1 number is that if they are not prime, they don't have to be the sum of two squares. For example, 21 cannot be written as the sum of two perfect squares, even though it is (4×5) + 1.