what is the probability that a prime number divides another prime plus 1?

what is the probability that a prime number divides another prime plus 1?

what I do know is that for 2 it's 100%

I can show this fact using a function

$$f(x,y):=$$ the number of primes between $$1$$ & $$y$$ that when you add 1 you can divide it by $$prime(x)$$ and get a whole number and divide that by $$π(y)$$

$$π(x)$$ is the prime counting function

$$f(1,x)=(π(x)-1)/π(x)$$ because the only time $$prime(x)+1$$ doesn't equal an even number is when the prime is $$2$$.

$$2+1$$ isn't even.

and as the x goes to infinity $$(π(x)-1)/π(x)$$ goes to 100%

my question is what is the probability that $$3,5,7,...$$ divides a random prime number plus 1

do you know the general formula for $$f(x,y)$$ as $$y$$ goes to infinity

• I can show that $2$ divides $p+1$ for all primes $p >2$ without using calculus. – Torsten Schoeneberg Jul 11 at 0:22
• yea I did go overboard all primes are odd beside 2 and odd plus 1 is, even so, I didn't need all that @TorstenSchoeneberg – BriggyT Jul 11 at 0:26

If you fix the first prime $$p$$ then you are looking for primes of the form $$pk-1$$ The density of this set of primes is $$1/(p-1)$$ in the set of all primes by the Chebotarev theorem.