# Number of primes that can be there in $k$ consecutive natural numbers

I want to know thenumber of primes that can be there in $$k$$ consecutive natural numbers. For example we can have 0,1 or 2 primes in 3 consecutive natural numbers.

Let $$N_{i,k}$$ be the number of primes in $$k$$ consecutive natural numbers starting from $$i$$.
We can easily prove that $$N_{i,k}=N_{i+1,k}$$ or $$N_{i,k}=N_{i+1,k}+1$$ or $$N_{i,k}=N_{i+1,k}-1$$. So we can conclude that when $$i$$ increased by $$1$$ the number of primes in $$k$$ consecutive natural numbers changes by atmost $$1$$.

It is easy to see that $$N_{2, k} \ge N_{1,k}$$ and $$N_{(k+1)!+1,k}=0$$. So we can say that when the value of $$i$$ in $$N_{i,k}$$ changes from $$2$$ to $$(k+1)!+1$$ then $$N_{j,k}$$ takes all natural values from $$0$$ to $$N_{2, k}$$ for some $$2\le j \le (k+1)! +1~~($$ because when $$i$$ increased by $$1$$ the number of primes in $$k$$ consecutive natural numbers changes by atmost $$1)$$

Even though I was able to prove the lower bound, I was not able to prove that the maximum value of $$N_{i, k}$$ is $$N_{2, k}$$. Any help to prove the upper bound would be appreciated.

You have rediscoveres a famous conjecture from analytic number theory! Let's translate to the standard notation $$\pi(x)$$ which denotes the number of primes less than or equal to $$x$$, so that $$N_{i,k} = \pi(i-1+k)-\pi(i-1)$$. Hardy and Littlewood conjectured that $$\pi(x+k)-\pi(x) \le \pi(k)$$ for all $$x,y\ge2$$.
Interestingly, it is now believed that this conjecture is false, as it is known to contradict the prime $$k$$-tuples conjecture. The refined conjecture, which is more likely to be true, would be that $$\pi(x+k)-\pi(x) \le 2\pi(\frac k2)$$. (In other words, the most primes in an interval of length $$k$$ should occur on the interval $$[-\frac k2,\frac k2]$$ give or take $$1$$, where we count negatives of primes as primes.)
• Small typo: you have $\pi(i-i)$ instead of $\pi(i-1)$. Do you have a reference for the refined conjecture? My understanding is that in Erdös’s time he wasn’t sure if the RHS could be as large as $\pi(k) + C k/(\log k)^2$ with $C$ arbitrarily large, but this hypothesizes a specific upper bound for $C$. Commented Jun 1, 2021 at 23:58
• @Asher2211 No change. Prime $k$-tuples asserts infinitely many constellations of an admissible form, so infinitely many would consist of only positive primes. Commented Jun 2, 2021 at 0:00