Schinzel-Sierpinski conjecture

Statement

The name Schinzel-Sierpinski conjecture is used for two related conjectures:

For any positive rational number $r$ , there exist prime numbers $p, q$ such that $r = \frac{p + 1}{q + 1}$ . A stronger version of the conjecture asserts that there are infinitely many such pairs of prime numbers.
For any positive rational number $r$ , there exist prime numbers $p, q$ such that $r = \frac{p - 1}{q - 1}$ . A stronger version of the conjecture asserts that there are infinitely many such pairs of prime numbers.