Motivated by questions such as this one, I have developed a proof of the following theorem, which I need to be subject to peer review:
Let $n,k$ be positive integers, and consider the interval $$ I = [kn, (k+1)n] = \{kn, kn+1, kn+2, \dots, (k+1)n\}. $$ Assume $(k+1)n < p_{\pi(n)+1}^2$, where $p_{\pi(n)+1}$ is the first prime greater than $n$. Then, the number of composite numbers in $I$ cannot exceed $n$. Therefore, at least one number in $I$ must be prime.
Proof
For each prime $p \leq n$, let: $$ A_p = \{x \in I \mid p \text{ divides } x\} $$ $$ B_p = \{x \in [0,n] \mid p \text{ divides } x\} $$ Lemma: $|A_p| \leq |B_p|$
Lemma proof
Any $x \in A_p$ has the form $x = kn + r$ with $kn \leq x \leq (k+1)n$ and $p \mid x$.
Define a function: $$ f \colon A_p \longrightarrow B_p, \quad f(kn + r) = r. $$
The function
(i) is well-defined: if $kn + r \in A_p$, then $r$ satisfies $r \equiv -kn \pmod{p}$ and $0 \leq r \leq n$, so $r \in B_p$.
(ii) It is injective: suppose $f(kn + r_1) = f(kn + r_2)$, meaning $r_1 = r_2$. Then, since $kn + r_1 = kn + r_2$, we have $x_1 = x_2$, so the function is one-to-one.
Hence $|A_p| \leq |B_p|$, as desired. $\square$
Using the Lemma, together with the fact that $$B_{p_1} \cap B_{p_2} \leftrightarrow A_{p_1} \cap A_{p_2}$$ we have that $$ \left|\bigcup_{p\leq n} A_p\right| \leq \left|\bigcup_{p \leq n} B_p\right| $$
Key fact: Notice that 1 is not divisible by any prime. The set $\bigcup_{p \leq n} B_p$ is exactly $[0,n]$ with $1$ removed, because $1$ has no prime divisors. Thus, $$ \bigcup_{p \leq n} \{y \in [0,n] : p \mid y \} = [0,n] \setminus \{1\}. $$
Thus, $\left|\bigcup_{p \leq n} B_p\right| = [0,n] \setminus \{1\}$, which has only $n$ elements. Therefore, by the Lemma and its corollary, we deduce that there can be at most $n$ composite numbers in the interval $I$. Since $I$ has $n+1$ distinct numbers, there must be at least one integer in $I$ which is not composite; such an integer must then be prime.
Hence we conclude that there is at least one prime in the interval $[kn,(k+1)n]$, completing the proof of the theorem.
$\square$
I have sent it to a top tier journal -as (if correct) the theorem directly implies several classical conjectures in number theory, including Oppermann's, Legendre's, Brocard's, and Andrica's conjectures, as well as new bounds on prime gaps- and I was told literally:
Because so many authors have submitted false solutions to the problem addressed in your manuscript, we can only consider such solutions if the exposition is exceptionally clear. If you are convinced that your solution is correct, and wish to continue to pursue publication, then you should have someone else (for instance a mathematically literate friend or colleague, or perhaps a mathematician at a local university) read your manuscript and give you suggestions for improving the readability. You should submit your manuscript again to a journal only if that person is able to understand your manuscript well enough to certify its correctness.
Therefore, I would appreciate any kind (or unkind) peer-review, suggestion, etc., that you could provide, as unfortunately I do not have any close colleague that can review it and certify / discard the validity of the theorem and its proof.
Thanks in advance for your help!
