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Results tagged with prime-numbers
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user 122662
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
3
votes
2
answers
339
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For any integer $n>1$, there always exists at least one prime number $p$ with $n < p< n+\lef...
Question: Is the conjecture as follows true or false?
For any integer $n>1$, there always exists at least one prime number $p$ with
$$n < p< n+\left(\ln\Big(\frac{n}{\ln n}\Big)+1\ri …
- 1,776
1
vote
1
answer
153
views
Is $P_{2n} \ge 2P_n$ and $\pi(2x) \le 2\pi(x)$ inconsistent to the Prime k-tuple conjecture?
I posed a conjecture as follows that is a special case of Second Hardy–Littlewood conjecture:
For $n, x \ge 2$ be two integers then:
$$P_{2n} \ge 2P_n$$
and
$$\pi(2x) \le 2\pi( …
- 1,776
2
votes
1
answer
339
views
Is there a constant $\alpha$ such that: $P_{n+1} < P_n.\left(\frac{n+1}{n}\right)^\alpha$?
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 …
- 1,776
6
votes
2
answers
711
views
Does always exist a prime number between $n(n+1)$ and $(n+1)(n+2)$?
Let $n$ is positive integer number, does always exist a prime number between $n(n+1)$ and $(n+1)(n+2)$?
- 1,776
4
votes
4
answers
894
views
Let $X$ be a positive integer. Then $\pi{(X+\ln^2{X})}-\pi{(X-\ln^2{X})}>\ln{X}$?
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 …
- 1,776
3
votes
0
answers
87
views
Are there infinitely many primes $p$, positive integers $ k, n $ such that $1 \le n < p$ and...
Among $168$ prime numbers in range $1$ to $10^3$, there are $84$ prime numbers $n$ such that: $p^k> n.rad(p^{k+1}−n)$ where $1 \le n<p$ and $k=2,3,4$. There are also $84$ prime numbers $n$ such that: …
- 1,776
-4
votes
1
answer
228
views
A generalization Bertrand's postulate [closed]
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 …
- 1,776
7
votes
1
answer
659
views
Prove: If $P_n$ is $n$-$th$ prime number then $P_{n+m} \ge P_n+P_m$
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?
- 1,776
17
votes
2
answers
2k
views
Is every odd positive integer of the form $P_{n+m}-P_n-P_m$?
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_{ …
- 1,776
8
votes
2
answers
385
views
Can exist a positive integer number $x$ such that $a_1=x$ and $a_n=2a_{n-1}+1$ are not prime...
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 …
- 1,776
6
votes
2
answers
1k
views
$\pi((n+1)^2)-\pi(n^2) \le \pi(n)$ for all $n \ge 370$?
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 …
- 1,776
4
votes
0
answers
334
views
The number of continuously increasing primes gaps in the interval $[2,n]$ is less than $\log n$
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 …
- 1,776
3
votes
1
answer
396
views
Probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$
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 $N=10^2, 10^3,...,10^{9}$, $x$ …
- 1,776