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

  • 2
    $\begingroup$ I can show that $2$ divides $p+1$ for all primes $p >2$ without using calculus. $\endgroup$ – Torsten Schoeneberg Jul 11 at 0:22
  • $\begingroup$ 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 $\endgroup$ – 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.

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  • $\begingroup$ thanks that makes sense $\endgroup$ – BriggyT Jul 11 at 0:24

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