MHB Discovering Intersections of Infinite Primes Sets

[caters]

We have this set of primes which is infinite. This has lots of different subsets. Here is the list of subsets:
Real Eisenstein primes: 3x + 2
Pythagorean primes: 4x + 1
Real Gaussian primes: 4x + 3
Landau primes: x^2 + 1
Central polygonal primes: x^2 - x + 1
Centered triangular primes: 1/2(3x^2 + 3x + 2)
Centered square primes: 1/2(4x^2 + 4x + 2)
Centered pentagonal primes: 1/2(5x^2 + 5x + 2)
Centered hexagonal primes: 1/2(6x^2 + 6x + 2)
Centered heptagonal primes: 1/2(7x^2 + 7x + 2)
Centered decagonal primes: 1/2(10x^2 + 10x + 2)
Cuban primes: 3x^2 + 6x + 4
Star Primes: 6x^2 - 6x + 1
Cubic primes: x^3 + 2
Wagstaff primes: 1/3(2^n + 1)
Mersennes: 2^x - 1
thabit primes: 3 * 2^x - 1
Cullen primes: x * 2^x + 1
Woodall primes: x * 2^x - 1
Double Mersennes: 1/2 * 2^2^x - 1
Fermat primes: 2^2^x + 1
Alternating Factorial Primes: if x! has x being odd than every odd number when you take the factorial positive and every even number negative. Opposite for even indexed factorials. For example 3rd alternating factorial = 1! - 2! + 3!
Primorial primes: First n primes multiplied together - 1
Euclid primes: first n primes multiplied together + 1
Factorial primes: x! + 1 or x! - 1
Leyland primes: m^n + n^m where m can be anything not negative but n has to be greater than 1
Pierpont primes: 2^m * 3^n + 1
Proth primes: n * 2^m + 1 where n < 2^m
Quartan primes: m^4 + n^4
Solinas primes: 2^m ± 2^n ± 1 where 0< n< m
Soundararajan primes: 1^1 + 2^2 ... n^n for any n
Three-square primes: l^2 + m^2 + n^2
Two Square Primes: m^2 + n^2
Twin Primes: x, x+2
Cousin primes: x, x+4
Sexy primes: x, x + 6
Prime triplets: x, x+2, x+6 or x, x+4, x+6
Prime Quadruplets: x, x+2, x+6, x+8
Titanic Primes: x > 10^999
Gigantic Primes: x > 10^9999
Megaprimes: x > 10^999999

Now Here is a question. Can you find a number where at least 2 of the sets intersect? I will try to do this myself. Just so you know I am going up to 12 digit primes because that is the largest prime my computer will test without the program taking too long to test it and to be sure I find intersections of the sets.

Another question is if I use the number of intersections between 2 sets of 2 types of primes up to 12 digits and I compare that to the number of intersections in a different pair of sets can I figure out which sets have the most intersections?

I am in 9th grade but know some trig and precalculus.

 

[Nono713]

Now Here is a question. Can you find a number where at least 2 of the sets intersect? I will try to do this myself. Just so you know I am going up to 12 digit primes because that is the largest prime my computer will test without the program taking too long to test it and to be sure I find intersections of the sets.

Well, yeah, it's trivial. 5 is a "real eisenstein prime", a pythagorean prime, a landau prime, a fermat prime, a factorial prime, it has two twin primes (3 and 7), and so on... there are infinitely many primes that are titanic primes, gigantic primes, and megaprimes (since all megaprimes are both titanic and gigantic, and there are infinitely many of them), etc... many of these sets overlap, some of them significantly (e.g. eisenstein and gaussian primes).

Another question is if I use the number of intersections between 2 sets of 2 types of primes up to 12 digits and I compare that to the number of intersections in a different pair of sets can I figure out which sets have the most intersections?

I don't know that an exhaustive test to check how much overlap any two sets have up to 12 digits is going to tell you very much about the behaviour of the intersection of the whole sets. Primes are highly irregular, it's not even known that a lot of these subsets are actually infinite. In fact I'm pretty sure for most of these subsets getting any (mathematical) upper bound on the number of intersections below a certain prime is going to be difficult, and a few could probably even be considered open problems.

But yes, if you restrict yourself to primes up to 12 digits, then you can work out the intersection of each set with every other, and then use the inclusion exclusion principle to figure out which sets intersect the others the most according to some metric of your choosing.