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Results tagged with prime-numbers
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user 37555
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.
17
votes
Accepted
Number of primes with $-1\pmod 6$ vs. Number of primes with $+1\pmod 6$
Assume that the Riemann hypothesis for the non-principal $L$-series $\pmod{3}$ is false, say, this series has a zero $\rho=\sigma+i\gamma$ with $\sigma>1/2$. Then Turan and Knapowski have shown that b …
6
votes
Accepted
Prime factors of the members of a certain recurrence
It is even true that the equation $n^2+1=m$ with $p|m\Rightarrow p\in S$ has only finitely many solutions. To see this note that every $m$ satisfying this property can be written as $m=m_1m_2^3$, wher …
15
votes
Accepted
Does this prime-gaps pattern occur infinitely often?
This would follow from the $k$-tuple conjecture in the following way.
Choose an admissible tuple $d_1, \ldots, d_{k+2}$, such that $d_2-d_1=d_{k+2}-d_2$. If $n\in[d_1, d_{k+2}]$ is an integer, such t …
3
votes
Best provable and unconditional lower and upper bounds for Brun's constant
The best known bounds seem to be due to Nicely [``A new error analysis for Brun's constant,'' Virginia J. Sci. 52 (2001), no. 1, 45–55), who showed that Brun's constant is
$$
1.9021605823 \pm 0.000000 …
1
vote
Minimal number of different values in the sequence $(\mu(d_i)\varphi(d_i))_{i=\overline{{1,\...
I assume that $m$ is squarefree, for otherwise the minimum would be equal to 2 no matter what $k$ is.
Let $p_1, \ldots, p_k$ be the set of all prime numbers of the form $2^a3^b+1$, where $a, b<t$. Th …
1
vote
Estimating $\sum_{p_1\cdots p_k\leq n} \frac{1}{p_1\cdots p_k}$ for various $k$
Define the function $F^*(n)=\underset{p_1\dots p_k\leq n}{\sum_{p_, \ldots, p_k}}\frac{1}{p_1\dots p_k}$. Then we have for each fixed $k$ the asymptotics $F_k^*(n)\sim(\log\log n)^k$. To see this note …
8
votes
Primality test for $2p+1$
The if-part is a special case of Pocklingtons theorem (see https://en.wikipedia.org/wiki/Pocklington_primality_test ). The only if part requires some computation, as you need a quadratic non-reside mo …
2
votes
Function with zeros plus/minus the primes
A standard method to improve the speed of convergence is to look for approximations, which can be explicitly evaluated. In this case one would take $f(s)\approx\zeta(2)^{-s^2}$. We have
$$
f(s)\zeta(2 …
1
vote
Accepted
Game: Avoid the Gaussian primes
This is only a heuristics, but I would assume that the game either ends after a few steps, or the outcome is determined simply by the parity of $(X+1)(Y+1)$ minus the number of gaussian primes in the …
6
votes
Quadratic residues and nonresidues of arbitrary patterns
The number of integers $a$ in $[0, x]$ with the desired property is
$$
2^{-n}\sum_{a=1}^x\prod_{i=1}^n\left(1+\epsilon_i\left(\frac{a}{p_i}\right)\right).
$$
Expand the right hand side to obtain one t …
1
vote
Writing integers as determinants of matrices with prime entries.
Goldston, Graham Pintz and Yildirim ( https://arxiv.org/abs/0803.2636 ) showed that among three linear forms $\ell_1, \ell_2, \ell_3$, which satisfy the obvious local conditions, there are two, say $\ …
2
votes
Accepted
Density of integers with a large rough divisor
If $a/b$ is not too large, you can compute the probability using arguments as in the computation leading to the asymptotics for smooth numbers. In theory, you can compute for all $\beta, \gamma$ a rea …
1
vote
Spacing of fractions with prime denominator
Wolke (On the large sieve with primes, Acta Math. Acad. Sci. Hungar. 22 (1971/72), 239–247, MathSciNet MR0291121 (45 #215)) has worked on this question. His motivation was Gallagher's approach to the …
10
votes
Is Li(x) the best possible approximation to the prime-counting function?
Whether for a finite set $\mathcal{R}$ of roots the approximation
$$
\pi(x)\approx\mathrm{Li}(x)-\frac{1}{2}\mathrm{Li}(x^{1/2})-\sum_{\rho\in\mathcal{R}}\mathrm{Li}(x^\rho)
$$
is "on average" better …
11
votes
1
answer
408
views
Integers with a large prime divisor in short intervals
For an integer $n$, denote by $P^+(n)$ the largest prime divisor of $n$. Then we have the following:
There exists some $c>0$, such that for all $x$ sufficiently large the number of integers $n\in[x, …