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 all except the powers of 10 ones intersect? Are there more intersections besides that? I will try to do this myself.