All Questions
130 questions
7
votes
0
answers
786
views
"Forthcoming paper" of Goldston-Graham-Pintz-Yıldırım
The above-named authors of [1] and its (significantly different) published version [2] write:
In a forthcoming paper, we will show how the methods here can be extended to prove corresponding ...
- 9,114
1
vote
2
answers
375
views
binomial/factorial identity mod p
In trying to determine the spectrum of a well-known ergodic transformation, I came up with the following useful (for me) result.
Let $p$ be a prime and $a$ a positive integer. Then for $M$ a positive ...
- 4,679
0
votes
3
answers
415
views
Has this formula for $G_{k}:=\lim\inf_{n\to\infty}p_{n+k}-p_{n}$ been conjectured?
I give here a heuristics that suggests that the quantity $\displaystyle{G_{k}:=\liminf_{n\to\infty}p_{n+k}-p_{n}}$ should be approximately equal to $k(1+H_{k})$, where $H_{k}$ is the $k$-th harmonic ...
- 6,984
11
votes
2
answers
1k
views
Most dense subset of numbers that avoids arbitrarily long arithmetic progressions
The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression.
I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid
...
- 151k
4
votes
0
answers
117
views
Best constant for Maier's theorem?
Maier proved that, for fixed $\lambda>1,$
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\pi(x)}{\log^{\lambda-1}x}>1
$$
and in particular
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\...
- 9,114
4
votes
1
answer
456
views
References to proofs of upper and lower bounds on the number of coprimes in an interval?
On the first page of the article "When the sieve works", the authors present upper and lower bounds for $S(T,T+x;\mathcal{E})$; the number of integers in the interval $(T,T+x]$ that are coprime to all ...
- 965
7
votes
1
answer
1k
views
What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?
The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log ...
- 6,984
6
votes
1
answer
367
views
Bounds re Asymptotic Formula for the Sum of Largest Prime Factors
I have a reference request related to the result :
$\sum_{n=2}^{x} P(n)$ ~ $\frac{\pi^2}{12}\frac{x^{2}}{log(x)}$ as $x \rightarrow \infty$
where $P(n)$ is the largest prime factor of the positive ...
- 71
7
votes
2
answers
1k
views
Is there a von Koch-type theorem for the generalized Riemann hypothesis?
Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the error term in the prime number theorem having the bound
$$
\mid\pi(x)-\textrm{li}(x)\mid=O(\sqrt{x} \log x).
$$
Q1: ...
- 965
11
votes
1
answer
408
views
Integers with a large prime divisor in short intervals
For an integer $n$, denote by $P^+(n)$ the largest prime divisor of $n$. Then we have the following:
There exists some $c>0$, such that for all $x$ sufficiently large the number of integers $n\in[...
9
votes
0
answers
414
views
In which orders can the numbers of prime factors of consecutive integers be?
Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur.
Given a ...
- 2,719
2
votes
0
answers
207
views
n-ary quadratic forms with $S$-integer values
Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form.
Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to $Q(x_1,\ldots,...
- 1,639
4
votes
2
answers
730
views
Looking for paper: Weil's original 1952 "Sur les formules explicites de la théorie des nombres premiers"
I am looking for a source (preferably online) for Weil's original 1952 paper on the explicit formula. I am aware of an english translation available here, but would like to have access to the original ...
- 3,799
7
votes
1
answer
435
views
Are primes of density 0 in $a\cdot b^n+c$?
Hooley proves in Applications of Sieves to the Theory of Numbers that there are only $o(x)$ numbers $n\le x$ such that $n\cdot2^n+1$ is a (Cullen) prime. The proof generalizes to forms $n\cdot2^{n+a}+...
- 9,114
7
votes
1
answer
652
views
Fermat-quotient of "order" 3: I found $68^{112} \equiv 1 \pmod {113^3}$ - are there bigger examples known?
(I've taken this from MSE, it seems to be more appropriate here)
I'm rereading an older text on fermat-quotients (see wikipedia) from which I have now the
Question for
$$ b^{p-1} \equiv 1 \pmod{ ...
- 5,342
16
votes
1
answer
4k
views
Order of magnitude of $\sum \frac{1}{\log{p}}$
Question: What is the order of magnitude of the following sum?
$$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$
Additional information: Since
$$ \sum_{\substack{p<n\\\text{...
- 3,004
11
votes
1
answer
1k
views
The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$
The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
a\...
- 11.4k
3
votes
1
answer
398
views
Primes in short intervals with a preassigned frobenius
Edited after mistake in the first version.
It is known since Selberg that under the Riemann Hypothesis, given an $\epsilon>0$, there is a prime between $x$ and $x+O(x^\epsilon)$ for all $x$ in a ...
- 26k
5
votes
1
answer
455
views
Large gaps between P2s
Gaps between consecutive primes are $O(n^{\theta+\varepsilon})$ for $\theta=0.525$ and any $\varepsilon>0.$ I was wondering if a better result is known for gaps between numbers with at most two ...
- 9,114
6
votes
1
answer
1k
views
Reference request: Dickman, On the frequency of numbers containing prime factors
I've been trying without success to find the paper
Dickman, Karl, "On the frequency of numbers containing prime factors of a certain relative magnitude." Ark Mal., Astronomi och Physik, 22A (10), ...
- 1,077
3
votes
3
answers
670
views
Proof of infinitude of primes whose reversal in base 10 is also prime [closed]
Is there any proof of infinitude of A007500 primes?
If you want to generate them here is trivial and naive python program.
...
- 481
4
votes
1
answer
928
views
Primes and Ackermann's function
If $A(m,n)$ is Ackermann's function, and $c$ is a fixed integer, are there any heuristics/conjectures/obvious things that can be said about primes of the form $A(m,n)+c$, $m \geq 4$,at all?
EDIT:
I ...
- 1,075
25
votes
2
answers
4k
views
Primes of the form $x^2+ny^2$ and congruences.
The answer of following classical problem is surely known, but I can't find a reference
For which positive integer $n$ is the set $S_n$ of primes of the form $x^2+n y^2$ ($x$, $y$ integers) ...
- 26k
6
votes
2
answers
754
views
ASCII prime plots and prime-rich quadratic polynomials
This is a series of questions inspired by the MathOverflow question
Find the least prime so that p-1 has two factors greater than $m$ and $n$ posted by Aaron Sterling.
I suggested plotting primes by ...
- 13k
1
vote
1
answer
453
views
Are large numbers the sum of two or more large primes? [Hoping for reasonable constants]
Is it true that for all $n>N$ that n is the sum of two or more distinct primes that are either large or (for parity reasons) 2?
I feel like I've seen a result allowing this with $p\gg n^e$ for ...
- 9,114
4
votes
0
answers
369
views
Reducing factoring prime products to factoring integer products (in average-case)
My question is about the equivalence of the security of various candidate one-way functions that can be constructed based on the hardness of factoring. (This question has been asked also in the CS ...
- 41
3
votes
2
answers
876
views
Asymptotics for the number of partitions of $n$ into odd prime parts
Hello!
I am interested in the asymptotic behavior of the function $p_o(n)$ defined as the number of partitions of $n$ into odd prime parts A099773 - http://oeis.org/A099773 .
I couldn't find any ...
- 3,463
18
votes
3
answers
6k
views
The multiplicative order of 2 modulo primes
Artin's Conjecture says that any positive integer, which is not a square, is a primitive root modulo infinitely many primes. Christopher Hooley gave in
Hooley, Christopher (1967). "On Artin's ...
- 25.5k
7
votes
4
answers
1k
views
Reference for the expected number of prime factors of n larger than n^alpha is -log alpha
Let $0 < \alpha < 1$ be a constant. The expected number of prime factors of a "random" integer near $n$ which are greater than $n^\alpha$ is $-\log \alpha$.
It's my understanding that (...
- 14k
21
votes
3
answers
3k
views
Twin Prime Conjecture Reference
I'm looking for a reference which has the first statement of the twin prime conjecture. According to wikipedia, nova, and several other quasi-reputable resources it is Euclid who first stated it, but ...
- 1,588