I've been considering something involving median numbers.
If an integer is directly in the middle of two integers, is it possible to accurately extrapolate what two it is between?
Can a prime be in the middle of two primes?
Would this sort of information have any applications to digital cryptography?
On
More interestingly all numbers (ie natural numbers) lie mid-way between two primes. And that includes prime numbers as well. The only proviso being that the number in question is 4 or greater, other wise there would not be any suitable primes smaller than it.
But as the target gets larger then it is in general likely to to lie between an increasing number of primes. And that applies to the primes themselves.
Take any number $n$ and it is between $n-1$ and $n+1$, and $n-2$ and $n+2$, and $n-3$ and $n+3$, and so on. Asking "which number is $n$ in the middle of" has infinitely many answers. There are no applications to cryptography (or anything else) because the question is not well-defined.
To answer your second question, an example of a prime lying in the middle of two primes is $5$, which lies midway between $3$ and $7$.