Different Types of Prime Numbers

How often do primes happen?

From	There are
1 – 100	25 primes
1 – 1000	168 primes
1 – 10,000	1,229 primes
1 – 100,000	9,592 primes
1 – 1,000,000	78,498 primes
1 – 10,000,000	664,579 primes
1 – 100,000,000	5,761,455 primes

Twin Primes are 2 different prime numbers that differ by just 2.

For example: 3 and 5

5 and 7

11 and 13

17 and 19

29 and 31

Symmetric Primes (Euler Primes) are prime numbers that are the same distance away from a given number.For Example, 3 and 11 are both 4 units away from the number 7. Hence 3 and 7 are symmetric primes. Notice that all twin primes must be symmetric primes. All integers above 3 seem to have a symmetric prime, but I am not aware of any proof. Below is a list of a few.

Symmetric Primes	Number
5 and 11 or 3 and 13	8
5 and 13 or 7 and 11	9
3 and 17 or 7 and 13	10
7 and 19 or 3 and 23	13
5 and 23 or 7 and 19	14
9 and 29 or 7 and 31	19

Wilson’s Theorem states that if p is a prime number, ^{(p – 1)!/}_p will always leave a remainder of p – 1.

For example, let p be the prime number 5. Then ^{(5 – 1)!/}₅ = some number with a remainder of 5 – 1.

Doing the math, we get ^4!/₅ which is ²⁴/₅ = 4 with a remainder of 5 – 1, which is 4.

Let’s try the prime number 11. (11- 1)!/11 should give us a remainder of 10.

(11 – 1)!/11 = 10!/11 = 3,628,800/11 = 329890 with a remainder of 10.

Relatively Prime numbers is a term used to refer to two numbers that do not share a common factor; hence, the GCF (Greatest Common Factor) of the numbers would be 1. For instance, 5 and 8 are relatively prime since the only common factor they share is 1.

Mersenne Primes are primes that are found by the expression 2^p -1 where p is a prime number. Below is a table of a few.

2^p – 1	Mersenne Primes
2^3 – 1	7
2^5 – 1	31
2^7 – 1	127
2^13 – 1	8191

An emirp (prime spelled backwards) is a prime number whose palindrome(writing the number backwards – like 456 changed to 654) is a prime. A list of a few emirp is listed below.

11 13 17 31 37 71 73 79 97 101 107 113