Big Prime Numbers

Prime Numbers have always fascinated me. Nothing was more out of this world to me than a big number that is resistant enough to not divide into anything smaller than it, a number that can't be broken up into equal groups. I wish to learn more about the mathematics involved in prime numbers, as properties of prime numbers are still a relatively unknown topic in number theory and mathematics. Though I'll need to learn more about other fields of mathematics first as a foundation before analyzing something that is so simple to comprehend, but so difficult to make conclusions for. For this reason, I am currently sticking with just FINDING large primes, rather than working on things like the twin prime conjecture, which is an unsolved problem about the gaps between prime numbers.

Primes can be categorized in many different ways. The reason why primes are categorized, is because testing a random odd number for it's primality is much less likely to turn up prime, than some expression that generates a specific list of primes.

For example, a Mersenne Prime (named after Marin Mersenne, a French Mathematician and Physicist) is a prime of the form 2ⁿ - 1 (essentially you multiply a bunch of 2's, then subtract 1). I am not actually going to invest time into finding these, as all of the ones that are 1000 to 10000 digits long have been found long ago, and I don't have the programming knowledge (or the patience, lol), to do it for the largest ones. Nine of the top ten largest primes are Mersenne Primes, the largest (as of August 2021), being 2⁸²⁵⁸⁹⁹³³-1, at over 24 million digits long. Crazy!

I'm more invested in finding Proth Primes, which are very similar. You multiply any number, let's say 19, by once again, a power of 2, then add 1. In this case, it would be 19×2ⁿ+1. People aren't invested into primes of this form, so I guess I'm claiming the spot before other people do I guess?

I also occasionally find near-repdigit primes. The term 'rep-digit' is shorthand for 'repeated digit', which is defined as exactly what it sounds like: a repeated digit! Examples of 'rep-digits' are 11, 555555, 666666666666666, and 99999999999999999. With the exception of repeated 1's (e.g. 11, 1111111111111111111), rep-digits are never prime. For example, 8888888 is not prime, because it is clearly divisible by 8 (and 1111111 for that matter). However, you can't guarantee that a number isn't prime if you just change one of the digits, such as the number 333331, a prime number. near-repdigit prime numbers are primes that have all the same digits, with just one digit that is different from the rest.

Finally, I'll mention Picture Primes, which are primes that when arranged in a certain grid format, looks like a picture of some sort.

Prime Numbers are numbers that can only evenly divide into themselves and 1. For example, the number 7 is prime because the only way it can be split up is into groups of 1, or groups of 7 (itself). 6, on the other hand, is not a prime number, because although you can also split 6 into groups of 1, or groups of 6 (itself), it can also be split into two groups of 3 or three groups of 2. A number is only prime if it only divides evenly into 1 and itself.

For anyone interested, here are the prime numbers up to 100:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97