Find at least ten different prime numbers $p$ such that $p + 6$ is also prime.

Question

How can I find the set of prime numbers without using the direct trial and error?

Answer 1

As of now, we don't have any magical mathematical object which generates a kth prime or hands you a prime on demand provided the given condition(s).

What we do is that we try to establish if the given number is prime or not!

Establishing the fact that a given number is a prime number is an area of research with clever algorithms but most of them are computational in nature, that is, they are computation heavy and evaluated using the help of computers.

At the soul of such algorithm is their attempts to reduce the search space so as to reduce the computations.

The best you can do is to reduce your search space/computation time

Generate A List of All Odd Numbers having difference of 6.

Then:

• Better approach: Find the prime numbers among them, by comparing it with ready-made list of prime numbers

• More approaches can be added if OP specifies what kind of computational means are available to them.