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Results tagged with prime-numbers
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user 16537
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.
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An elementary proof that, for every fixed $n \in \mathbf N^+$, there are infinitely many pri...
This morning, I made a comment to a comment to a question of Ayman Moussa, only to point out that, among many others, there is an elementary proof of Dirichlet's theorem on the existence of infinitely …
- 10.5k
5
votes
Accepted
How composite $a^n+b$ is?
This is an answer to the part of the question where the OP is asking if the image of the function
$$R_{2,\varphi}: \mathbf N^+ \to \mathbf R: n \mapsto \frac{\varphi(2^n - 1)}{2^n-1}$$ is dense in th …
- 10.5k
5
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Is there some example that nicely extends the multiplication of natural numbers?
Edit (following the clarifications of the OP): This is not an answer, it is rather a long comment.
Studying this kind of questions is part of the mission of factorization theory: The language of the t …
- 10.5k
2
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What is the simplest proof that the density of primes goes to zero?
Terry Tao's proof can be slightly reworked to give a little more — that $\mu^\ast(P) = 0$ for every arithmetic upper quasi-density $\mu^\ast$ on $\mathbb N$, where $P$ is the set of primes. And a refi …
- 10.5k
0
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binomial/factorial identity mod p
As for references, you may want to give a look at the introduction and Section 2.2 of R. Meštrović's survey/preprint on Lucas's theorem (on arXiv).
- 10.5k
0
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Is there some example that nicely extends the multiplication of natural numbers?
[I'm adding a new (non-)answer since what follows has almost nothing to do with my previous one.]
Fix a real number $\alpha \ge 1$ and set $S_\alpha := \mathbb N \cup \mathbb R_{\ge \alpha}$. Of cours …
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