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I am looking for a comment, reference, remark, or proof of three conjectures as follows: Conjecture 1: Let $x$ be an odd positive integer. Then there exist two integers $n, m \ge 2$ so that $$x=P_{ …

Given a positive integer $P>1$, let its prime factorization be written as$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}.$$ Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k). …

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$ Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$ I …

Using my computer, I found that the most of positive integer number $x$ such that $a_1=x$ and $a_n=2a_{n-1}+1$ is prime number after a few iterations. But exist some positive integer numbers, my comp …

Let $n > 1$ and $m > 0$ be two integers and $P_n$ be the $n^{th}$ prime. Prove: $$P_{n+m} \ge P_n + P_m .$$ Can you give a hint, reference, comment, or proof?

Let $n$ is positive integer number, does always exist a prime number between $n(n+1)$ and $(n+1)(n+2)$?

There are some conjectures of the form: There always exist at least $X$ prime numbers between $A$ and $B$. Examples: Bertrand's postulate: for every $n>1$ there is always at least one prime $p$ such …

Let $P(n)$ be the statement that $$n < \mathrm{rad}(n(n-1)(n-2)),$$ where $\mathrm{rad}$ is the radical of an integer, that is defined as $$\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ pri …

A prime gap is the difference between two successive prime numbers. The $n$-th prime gap, denoted $g_n$ or $g(p_n)$ is the difference between the $(n+1)$-st and the $n$-th prime numbers. Using my comp …

The prime-counting function is the function counting the number of prime numbers less than or equal to some real number $x$. It is denoted by $\pi{(x)}$. Using my computer I found that for any posit …

Using my computer, I found that in the interval $[1, N]$ the probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$ is greater than constant $c$ where $N=10^2, 10^3,...,10^{9}$, $x$ …

Is there a constant $\alpha$ such that: $$P_{n+1} < P_n \cdot \left(\frac{n+1}{n}\right)^\alpha$$ Or $$\lim_{n\to\infty}\frac{\ln\frac{P_{n+1}}{P_n}}{\ln\frac{n+1}{n}} < +\infty$$ Whe …

I posed a conjecture as follows that is a special case of Second Hardy–Littlewood conjecture: For $n, x \ge 2$ be two integers then: $$P_{2n} \ge 2P_n$$ and $$\pi(2x) \le 2\pi( …

Question: Are the properties as follows holds? Version 1: the answer by Bjørn Kjos-Hanssen Let $P$ be a positive integers. We written: $P=$ $a_1^{x_1}a_2^{x_2}...a_n^{x_n}$ $=b_1^{y_1}b_ …

Let $n, k$ are integers number such that $1<n \le k$, does always exist a prime number between $kn$ and $k(n+1)$? When $n=1, k>1$ always exist a prime number between $k$ and $2k$ the question was pr …