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Results tagged with prime-numbers
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user 5101
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.
5
votes
Accepted
Inert primes in arithmetic progression
This is not true in general. Take $K = \mathbb{Q}(i), a =3$ and $m = 4$. Then the density is just $1/2$ in this case. This is because a prime $p$ is inert in $K$ if and only if $p \equiv 3 \bmod 4$. S …
- 21.4k
7
votes
2
answers
574
views
A variant on Wieferich primes
Recall that a Wieferich prime is a prime number $p$ such that
$2^{p-1} \equiv 1 \bmod p^2.$
It is not known whether there are infinitely many Wieferich primes, nor whether there are infinitely many no …
- 21.4k
6
votes
"Inner product" between prime factorizations
Here is a high-brow way to interpret your construction.
Recall that the ideles $\mathbb{A}^\times$ of $\mathbb{Q}$ are defined to be the restricted direct product of $\mathbb{R}^\times$ and all $\mat …
- 21.4k
10
votes
Accepted
Is there a Chebotarev‘s theorem for non-Galois extension over Q？
Actually the usual Chebotarev density theorem in the Galois case can also be applied to the non-Galois case.
For example, consider a non-Galois cubic extension $K=\mathbb{Q}[x]/(f)$. I claim that the …
- 21.4k
13
votes
4
answers
2k
views
Proving Mertens' theorem using the prime number theorem
Mertens' Theorem states that
$$\sum_{p \leq x}\frac{1}{p} = \log \log x + M + O(1/\log x).$$
This is weaker than the prime number theorem; in fact according to the Wikipedia page, the prime number the …
- 21.4k
12
votes
1
answer
521
views
Equidistribution of $\{\alpha p\}$ for $p$ in an arithmetic progression
Let $\alpha$ be irrational. A famous theorem of Vinogradov says that $\{ \alpha p\}$ is equidistributed in $[0,1]$ as $p$ runs over all primes.
Let $a,q$ be natural numbers with $\gcd(a,q) = 1$. Then …
- 21.4k