# Tagged Questions

Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

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### Number of primes of a certain form

Let $p_n$ be the nth prime. Are there an infinite number of primes of the form $2p_n+1$? Is something known about questions like this?
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### Linear convex combinations of $Li(x)=\int_2^x\frac{1}{\log(t)}dt$ and $\frac{x}{\log(x)}$, and prime counting function

Can provide us a linear convex combination of $Li(x)=\int_2^x\frac{1}{\log(t)}dt$ and $\frac{x}{\log(x)}$ a better approximation for $\pi(x)$, the prime counting function? Or not, is better $Li(x)$ ...
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### Do we have more primes of the form $3k+1$ or of the form $3k+2$?

Let us denote by $a_{1}(n)$ the number of prime numbers in the set $\{1,2,...,n\}$ which are of the form $3k+1$. Let us denote by $a_2(n)$ the number of prime numbers in the set $\{1,2,...,n\}$ which ...
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### Bounds on twin prime counting function

I read somewhere (unfortunately I cannot find the paper again) that the twin prime counting function $\pi_2(x)$ satisfies $\pi_2(x) \leq C\frac{x}{\log^2x}$ for some constant $C$. How would one prove (...
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### Can it be proven that infinite many primes can be formed only using two distinct digits?

It seems obvious that infinite many primes can be formed only using two distinct digits. $67776767776667777777$ is an example for such a prime. Even if we allow only the digits $0$ and $1$, there ...
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### Can every odd prime $p\ne 11$ be the smallest prime factor of a carmichael-number with $3$ prime factors?

According to my search, the number $561=3\times11\times17$ is the only carmichael-number with $3$ prime factors, which is divisible by $11$. Is this true ? If yes, $11$ cannot be the smallest ...
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### Asymptotic local limit theorem and applications in analytic number theory

I'm wondering if one could get similar results to the classical local limit theorem if one assumes that conditions, such as independence and identicallity of distribution of the random variables ...
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### Representing integers as difference of semi-primes

Can every integer, $k$ (where $4 \le k < n$), be written as the difference of two semi-primes whose prime factors are at most $n$ ? or put another way: Given positive integers $k\ge 4$, $n > k$...
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### Some questions about $d$-primes

Here : https://oeis.org/search?q=3%2C7%2C13%2C31%2C103%2C109%2C151&sort=&language=&go=Search the primes are mentioned which remain prime if the digit $1$ is inserted at any position, ...
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### Power series with products of prime numbers as coefficients

I encountered in my work an infinite series (only even power) with $n^{th}$ coefficient being of the following form: $$\frac{1}{3\cdot5\cdot7\cdot11\cdot...\cdot(n^{th} prime)}.$$ I wonder if such ...
### Which odd composite numbers $n$ are strong-pseudoprime to no base $a$ with $1<a<n-1$?
In this question : A composite odd number, not being a power of $3$, is a fermat-pseudoprime to some base I did not hit my own intend. I only asked for the numbers $n$ that are not a fermat-...