# 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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### Primes in the binomial transform of $[1, 1, 2, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, …]$.

This question is related to this sequence A139482. A commentator gives the following formula for $a_m$ $$a_m = {3m^2-9m+10 \above 1.5pt 2}$$ I have that you should consider the sequence $b_n =3n+2$ ...
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### Is a tight concrete bound for the error-term in the prime-number-theorem known?

Here : https://en.wikipedia.org/wiki/Prime_number_theorem it is mentioned that $$\pi(x)=Li(x)+O(xe^{-a\sqrt{ln(x)}})$$ What is a tight upper bound for $|\pi(x)-Li(x)|$ in concrete terms ? The ...
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### Generalization of Inkeri's primality test

How to prove that following hypothesis is true ? Definition Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ , where $m$ and $x$ are ...
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### Longest sequence of primes where each term is obtained by appending a new digit to the previous term

What is the longest known sequence of primes where each new term is obtained by appending a new decimal digit to the previous term? Examples: $$(2,23,233,2333,23333)$$ There are no more members in ...
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### How to test if $n!+1$ is prime or not?

for $n=0,1,2,3,11,27,37,41,73,77,116,154,320,340,399,427,872,1477,6380,26951,...$ $$n!+1$$ is prime. But how can you proof (with 100% certantiy) thats the case? Especially for the larger ones. For ...
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### For every prime $p > 3$ that is $3$ mod $4$, does $q+1 \mid p-q$ for some other prime $q$?

Yet another random conjecture about primes: Given a prime $p>3$ of the form $4n+3$. Then there exist a prime $q<p$ such that $q+1\mid p-q$. Verified for all $p<100000$.
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### Numbers $a$ such that if $a \mid b^2$ then $a \mid b$

I want to describe the set of numbers $a$ such that if $a \mid b^2$ then $a | b$ for all positive integers b using the prime factorizations of $a$ and $b$. What would be a good way to approach this ...
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### Asymptotics of $\sum\limits_{n/2 < p \leq n} \frac{1}{p}$
I'm reading a paper which asserts the following: $$\sum_{n/2 < p \leq n} \frac{1}{p} \sim \frac{\log 2}{\log n}$$ follows from prime number theorem, where the sum is taken over $p$ prime. What is ...