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Results tagged with prime-numbers
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user 31469
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.
8
votes
Reference request: Numbers composed of given primes
As Mikhail Tikhomirov commented, the denominator should have $\log$ of $p_j$ instead of $p_j$ itself.
For a reference see Theorem III.5.3, and more generally, section 2 (`The geometric method') of Cha …
- 14.6k
20
votes
Accepted
Prove $4\sum_{k=1}^{p-1}\frac{(-1)^k}{k^2}\equiv 3\sum_{k=1}^{p-1}\frac{1}{k^2}\pmod{p^2}$
This goes way back to Emma Lehmer, see her elementary paper on Fermat quotients and Bernoulli numbers from 1938.
Assume $p\ge 7$. First, I reformulate your congruence. I replace
$$4\sum_{k=1}^{p-1}\fr …
- 14.6k
7
votes
The Boolean algebra generated by sets of prime divisors of the numbers $2^n-1$
No. The gcd (greatest common divisor) of $\{ 2^{n_i}-1\}_{i=1}^{k}$ can be seen to be $2^{\gcd(n_1,\ldots,n_k)}-1$, by an application of the Euclidean algorithm. If $p$ divides this expression, then $ …
- 14.6k
2
votes
Probabilistic interpretation of prime number theorem
Another way to obtain OP's differential equation is as follows:
$$f(x) \approx \prod_{p < x} \left( 1-1/p \right) \approx \exp( - \sum_{p<x} 1/p) \approx \exp( -\int_{1}^{x} \frac{f(t) dt}{t}).$$
Taki …
- 14.6k
14
votes
Accepted
For any integer $n>6$, does there always exist a prime $p>n+1$ such that $p\mid 2^n-1$?
Yes (except for $n=12$, as noted by Gerry Myerson). This is due to A. Schinzel, in strong form, see "On primitive prime factors of $a^n−b^n$" (Proc. Camb. Philos. Soc. 58, 555-562 (1962)).
Let $P(n)$ …
- 14.6k
47
votes
Accepted
Are there any Fibonacci numbers that are sandwiched between twin primes?
(In collaboration with Z. Chase.)
A Fibonacci number $F_{n}$ is never sandwiched between two twin primes $(p,p+2)$.
This is because this would require $F_{n}+1$ to be a prime, but that can only happen …
- 14.6k
4
votes
Largest power of $p$ which divides $F_p=\binom{p^{n+1}}{p^n}-\binom{p^{n}}{p^{n-1}}$
Although the question was completely answered by others, I want to provide some more input.
An illuminating $p$-adic point of view is given in Section 7.1.6, "The Kazandzidis Congruences", of the (e …
- 14.6k
12
votes
Accepted
$\pi(x+200)-\pi(x)\leq 50$?
Yes.
Up to $207$ there are $46$ primes. Hence, the inequality is true for $x \le 7$.
Let $$\pi_{210}(x) = \textrm{card}(\{n \in [0,x] \cap \mathbb{N}, \, \gcd(n,210)=1\}).$$
For $x>7$, $\pi(x+200)-\pi …
- 14.6k
23
votes
Who first proved the generalization of Bertrand's postulate to (2n,3n) and (3n,4n)?
EDIT: Jose Brox, in another answer, had provided references which not noly predate Nagura's result, but they are also stronger.
J. Nagura, already in 1952, proved that the interval $(x,\frac{6}{5}x …
- 14.6k
1
vote
Weak Siegel–Walfisz property
There are examples of real-valued functions $f$ for which this property fails even for bounded $q$. For instance, given a function $f$ let
$$s_x = \sum_{n \le x,\, n \equiv 1 \bmod 3}f(n),$$
$$ t_x = …
- 14.6k
10
votes
Accepted
Average value of the prime omega function $\Omega$ on predecessors of prime powers
Yes, this is true.
First, let us observe that replacing prime powers with primes cannot make a difference, and the same goes to replacing $\Omega$ with $\omega$.
Erdős, in "On the normal number of pri …
- 14.6k
4
votes
Accepted
Prime omega function values on a product of prime powers predecessors
Let $\tau(n)=\sum_{d\mid n}1$.
H. Halberstam studied related problems in "On the distribution of additive number-theoretic functions. III.", J. Lond. Math. Soc. 31, 14-27 (1956). Given an irreducible …
- 14.6k
44
votes
Accepted
What is the simplest proof that the density of coprime pairs does not go to zero?
I would say that the standard proof is elementary enough, but here is an argument that avoids the Möbius function and the Riemann zeta function.
Let $A_d$ be the set of pairs $(a,b)$ where $a,b\le x$ …
- 14.6k
5
votes
Accepted
$\{ x/p\} $ on average
In short, there is cancellation in this sum when $w=o(x)$, but not when $w \asymp x$.
The intuition for the transition at $w\asymp x$ is this: the cancellation in $\sum_{p \le w}(\{x/p\}-1/2)$ comes f …
- 14.6k
4
votes
Asymptotic density of k-almost primes
The purpose of this answer is to give references to past works where the approach outlined by Lucia was carried out, especially a paper of Ramachandra.
Let $\alpha\colon \mathbb{N}\to \mathbb{C}$ be a …
- 14.6k