First forty primes with complex factors

Prime	Complex Factor 1	Complex Factor 2
2	1+i	1-i
5	2+i	2-i
13	3+2i	3-2i
17	4+i	4-i
29	5+2i	5-2i
37	6+i	6-i
41	5+4i	5-4i
53	7+2i	7-2i
61	6+5i	6-5i
73	8+3i	8-3i
89	8+5i	8-5i
97	9+4i	9-4i
101	10+i	10-i
109	10+3i	10-3i
113	8+7i	8-7i
137	11+4i	11-4i
149	10+7i	10-7i
157	11+6i	11-6i
173	13+2i	13-2i
181	10+9i	10-9i
193	12+7i	12-7i
197	14+i	14-i
229	15+2i	15-2i
233	13+8i	13-8i
241	15+4i	15-4i
257	16+i	16-i
269	13+10i	13-10i
277	14+9i	14-9i
281	16+5i	16-5i
293	17+2i	17-2i
313	12+12i	12-12i
317	14+11i	14-11i
337	9+16i	9-16i
349	18+5i	18-5i
353	17+8i	17-8i
373	18+7i	18-7i
389	17+10i	17-10i
397	19+6i	19-6i
401	20+i	20-i
409	120+3i	20-3i

Primes that have factors in the complex plane

If you are confortable with mathematics (especially Prime Numbers, Real Numbers, Complex Numbers and Fermat's Two Square Theorem), you can jump to the Conclusion

Fermat's Two Square Theroem

An odd prime number p can be expressed as a sum of two squares if and only if p ≡ 1 (mod 4) ie one more than a number divisible by 4.

Here are some examples:

5 = (4 X 1) + 1 = 2² + 1²
13 = (4 X 3) + 1 = 3² + 2²
17 = (4 X 4) +1 = 4² + 1²

These p ≡ 1 (mod 4) primes are not the only prime numbers the rest are of the form p ≡ 3 (mod 4) ie three more than a number divisible by 4.

Here are some examples:

7 , 11 , 19 , 23 These p ≡ 3 (mod 4) cannot be expressed as the sum of two squares.

Proofs of the theorem are very complex as shown at this link

Here is a video that includes a visual explanation which is more easy to follow though still somewhat complex:

The Real Numbers

Real numbers are broadly split into two main groups:

Rational Numbers

Numbers that can be written as a simple fraction ^a / _b (where b is not equal to 0). This includes natural numbers (1, 2, 3 ...) , whole numbers (0, 1, 2, 3 ... ), integers (positive/negative whole numbers), terminating decimals (e.g., (0.75), and repeating decimals (e.g., (0.333...).

Irrational Numbers

Numbers that cannot be expressed as a fraction. They feature non-terminating, non-repeating decimal expansions. Common examples include π (Pi) Euler's number e , and non-perfect square roots (like √2)

Real numbers can be thought of as all points on a number line.

Here is a link to a more complete description of Real Numbers. 

Some Multiplication Rules for Real Numbers

A positive number multiplied by a positive number equals a positive number
For example: +5 X + 2 = + 10

A positive number multipleid by a negative number equals a negative number
For example: +5 X -2 = -10

A negative number multiplied by a negative number equals a positive number
For example: -5 X -2 = +10

A Personal Aside

I remember the class in year 8 when my maths teacher taught us those multiplication rules.

I thought at the time that these rules were just agreed to by mathematicians. Now I would now use the term conventions. It was only much later that I realised that there were mathematical proofs for them.

The first two are fairly obvious but the third is not as clear.

Here is a video explaining why a positive times a negative is negative and a negative times a negative is positive.

The video at this link  provides four proofs for a negative X a negative = a positive.

A consequence of the third rule (a negative X a negative = a positive) is that the square root of a negative number does not exist - though as we will see later that should be the square root of a negative number does not exist in the Real Numbers.

Before learning the rules above it might seem that √-4 = -2, but we now know that -2 X -2 equals +4.

Imaginary Numbers

In the mid 16th century, Geralomo Cardono, while working on cubic equations came across solutions that contained the square root of a negative number - √-121

Rafael Bombelli decided to manipulate square roots of negative numbers using the same rules as for real numbers and found that doing so produced accurate real number results.

Mathematicians came to call these types of numbers Imaginary Numbers.

Leonhard Euler, in the 18th Century used the letter i to denote the imaginary unit √-1.

The definition of i is i² = -1 .

Complex Numbers

The development of the idea of the Complex Plane put imaginary numbers on a solid footing. The idea of the Complex Plane was not fully developed until the 19th Century. The video below A (Very) Brief History of the Complex Plane describes the development of ideas leading the the idea of the Complex Plane.

The diagram below shows the complex plane. It consists of two axes - the horizontal axis is the Real Numbers and the vertical axis the Imaginary Numbers.

Complex Numbers are written in the form a + bi, where a and b are real numbers. On the Complex Plane diagram below two complex numbers are shown: 3 + 2i and 3 - 2i . Real numbers are included in the Complex Numbers - they are the numbers on the horizontal axis, ie the Complex Numbers with an imaginary component of 0.

Complex Conjugates

The conjugate of a Complex Number is that number with the sign swapped
ie a + bi and a - bi. Some examples are: 3 + 2i and 3 - 2i , 2 + i and 2 - i ,
5 + 2i and 5 - 2i.

When Complex Conjugates are multiplied together the imaginary parts cancel leaving a Real Number.

(a + bi)(a - bi)
= a² - abi + abi - (bi)²
= a² - (b² i²)
= a² - (-1)(b²)     Note: i² = -1
= a² - (-b²)
= a² + b²

Conclusion

So Complex Conjugates multiplied together produce a Real Number that is the sum of two squares.

From Fermat's Two Square Theroem we know that an odd prime number p can be expressed as a sum of two squares if and only if p ≡ 1 (mod 4).

So complex conjugates that multiply together to produce a p ≡ 1 (mod 4) prime are complex factors of that prime number.

Here is a link to the first forty prime numbers with complex factors

Here is the video that first introduced me to the idea that some primes have complex factors. The video below is the section from that video that provides a geometric explanation why p ≡ 1 (mod 4) primes have complex factors and p ≡ 3 (mod 4) primes do not have complex factors.

Here is the full video which is well worth watching.