TITLE: bound of the number of the primes on an interval of length n
QUESTION [5 upvotes]: I made this observation and it seems reasonable to me to ask :if $n$ is a natural number then the number of the primes less than or equal to $n$ is denoted by $π(n)$ . is that true that in any interval of length $n$ there are at most $π(n)+1$ primes?(the $+1$ is needed for the trivial occasion where $n=p-1$ and the interval of length $n$ is $[2,p]$)
Alternative we can say that in any interval of length $n-1$ there are at most $π(n)$ primes.

REPLY [10 votes]: This is a well-known conjecture. It even has a name: the Second Hardy-Littlewood Conjecture, in the form: $\pi(x+y) \le \pi(x)+\pi(y)$ for $x, y \ge 2$.
For a long time, this was generally thought to be true. Then in 1974, Ian Richards showed that it was incompatible with the First Hardy-Littlewood Conjecture! He did this by constructing explicitly an admissible prime constellation of length $x$ and size larger than $\pi(x)$. Computers were involved. See here for more details.
The First H-L Conjecture is considered a sure thing, which has led most mathematicions to abandon the Second H-L Conjecture (although any counterexamples are likely to be extremely large).

REPLY [2 votes]: This is indeed a known open problem,  the Hardy-Littlewood conjecture:
$$\pi(x+y) - \pi(x) \le \pi(y)$$