bound of the number of the primes on an interval of length n

Question

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.

Answer 1

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).

Answer 2

A combination of the Brun-Titchmarsh inequality and the Prime Number Theorem will yield the following: For every $\epsilon>0$ there exists $N$ such that for $y>N$ and for every $x>0$ we have $$\pi(x+y)-\pi(x)<(2+\epsilon)\pi(y).$$

However, this is not quite as good as what you are asking, since you want for every $M,N$ $$\pi(M+N)-\pi(N)\leq \pi(M)$$ Is this true or not? According to my analytic number theory text (Montgomery and Vaughn):

It was once conjectured that $$\pi (M+N)\leq \pi (M)+\pi (N)$$ for $M>1$, $N>1$, but there is now serious doubt as to the validity of this inequality. Indeed, it seems likely that $\rho(y)>\pi (y)$ for all large $y$.

Here, $\rho(y)$ is defined as $$\limsup_{x\rightarrow\infty}(\pi (x+y)-\pi (x)).$$

Hope that helps.

Answer 3

This is indeed a known open problem, the Hardy-Littlewood conjecture:

$$\pi(x+y) - \pi(x) \le \pi(y)$$