Hey caters.
You should probably start by reframing your problem in terms of a mathematics problem with constraints. Remember that the first digit is x (MOD 10) the second digit is [x (MOD 100)] - x (MOD 10) and so on for other digits.
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.
Hey caters.
You should probably start by reframing your problem in terms of a mathematics problem with constraints. Remember that the first digit is x (MOD 10) the second digit is [x (MOD 100)] - x (MOD 10) and so on for other digits.