Questions tagged [prime-numbers]
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.
7 questions from the last 7 days
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On sums of a prime and a central binomial coefficient
Let $\mathbb Z^+$ be the set of positive integers. In 1934, Romanoff proved that
$$\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n+1=p+2^k\ \text{for some prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x>0.$$
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- 18k
1
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148
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On even numbers of the form $p+p'+2^k$
Let $\mathbb Z^+$ be the set of positive integers. In 1934, Romanoff proved that
$$\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n+1=p+2^k\ \text{for some prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x>0.$$
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- 18k
-1
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1
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Whether $2n>10$ can be written as $p+p'+2^a+2^b$ with $p$ and $p'$ consecutive primes?
In a paper published in 1971, R. Crocker proved that there are infinitely many positive odd numbers not of the form $p+2^a+2^b$ with $p$ prime and $a,b\in\mathbb Z^+=\{1,2,3,\ldots\}$. The proof makes ...
- 18k
3
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1
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Sum of prime divisors functions
I was idly thinking today about the functions $\displaystyle f(n) = \sum_{p \mid n} p$ and $\displaystyle F(n) = \sum_{p^e \| n} ep$, respectively the "sum of prime divisors" function and ...
- 101
6
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0
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144
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Status of Mills' constant
There is a rather confusing state of affairs at Wikipedia concerning Mills' constant. The article on formula for primes mentions that It is not known whether it is irrational, but the article on Mills'...
4
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0
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117
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Character and exponential sums over primes under GRH
Theorem 18.13 of Montgomery and Vaughan's multiplicative number theory book says (ignoring $\log $'s) $$\int _{-\delta }^{\delta }|\psi _\chi (\beta )|^2d\beta \ll \delta x\hspace {15mm}\psi _\chi (\...
- 1,656
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Are 6n ± 1 or more general arithmetic progression forms used as a core tool in advanced Number Theory proofs, beyond basic sieving? [closed]
Is there any research that uses the 6n ± 1 form or the more general form kn ± r to prove more important prime number theorems? (e.g., linked to Dirichlet's theorem on arithmetic progressions)
