All Questions
130 questions
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Primes of the form $2^{m_0}p_1^{m_1}\ldots p_r^{m_r}+1$
Is it known any example of a set of primes $\{p_1,\ldots,p_r\}$ with the following property: there are infinitely many $(m_0,\ldots,m_r)\in\mathbb N^{r+1}$ such that $2^{m_0}p_1^{m_1}\ldots p_r^{m_r}+...
- 509
1
vote
1
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247
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When is $a^{2^n}+1$ prime finitely often unconditionally?
Define generalized Fermat numbers following OEIS and mathworld.
For natural $a,n$ and $a$ even, the generalized Fermat number (GFN) is
$F_n(a)=a^{2^n}+1$.
Very large GFN primes are known (in the ...
- 3,866
4
votes
1
answer
339
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What is known about the largest prime divisor of the product of $k$ consecutive integers?
Take $k$ consecutive composite integers from a prime gap. What is known about the largest prime divisor of their product?
It seems to me that except for the triplet $(8,9,10)$ and the pair $(8,9)$ , ...
CommunityBot
- 1
10
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3
answers
1k
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Quantitative and elementary proofs of the Prime Number Theorem
I would like to know two things: one, whether the best quantative bounds in the Prime Number Theorem are still basically those given by the Vinogradov-Korobov zero-free region? and two, whether there ...
CommunityBot
- 1
0
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1
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714
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What is wrong with this counterexample to primality test assuming GRH? [closed]
From SMOOTH NUMBERS: COMPUTATIONAL NUMBER THEORY AND BEYOND Andrew Granville pp.13-14:
2j. Lenstra’s polynomial time test as to whether an integer that is conjecturally prime, is rigorously ...
- 25.4k
6
votes
1
answer
367
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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
5
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0
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425
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Conjectured new primality test for Mersenne numbers
How to prove that this conjecture about a new primality test for Mersenne numbers is true ?
Definition: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$
...
- 34.3k
11
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2
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1k
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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
...
- 6,523
1
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1
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601
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reference on Dirichlet theorem on primes in arithmetic progression
I appreciate if you could help me to find a reference (and a proof).
Combining Dirichlet theorem on primes in arithmetic progression with Chebotarev densitiy theorem, we know that given two positive ...
- 105k
7
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0
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786
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"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
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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 ...
- 10.5k
0
votes
3
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415
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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 ...
CommunityBot
- 1
4
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0
answers
117
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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
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\...
- 9,114
7
votes
1
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1k
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What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?
The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log ...
CommunityBot
- 1
4
votes
1
answer
456
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References to proofs of upper and lower bounds on the number of coprimes in an interval?
On the first page of the article "When the sieve works", the authors present upper and lower bounds for $S(T,T+x;\mathcal{E})$; the number of integers in the interval $(T,T+x]$ that are coprime to all ...
- 965
7
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2
answers
1k
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Is there a von Koch-type theorem for the generalized Riemann hypothesis?
Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the error term in the prime number theorem having the bound
$$
\mid\pi(x)-\textrm{li}(x)\mid=O(\sqrt{x} \log x).
$$
Q1: ...
- 105k
11
votes
1
answer
408
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Integers with a large prime divisor in short intervals
For an integer $n$, denote by $P^+(n)$ the largest prime divisor of $n$. Then we have the following:
There exists some $c>0$, such that for all $x$ sufficiently large the number of integers $n\in[...
- 43.7k
9
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0
answers
414
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In which orders can the numbers of prime factors of consecutive integers be?
Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur.
Given a ...
- 2,719
2
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0
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207
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n-ary quadratic forms with $S$-integer values
Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form.
Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to $Q(x_1,\ldots,...
- 1,639
21
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3
answers
3k
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Twin Prime Conjecture Reference
I'm looking for a reference which has the first statement of the twin prime conjecture. According to wikipedia, nova, and several other quasi-reputable resources it is Euclid who first stated it, but ...
- 9,114
11
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1
answer
1k
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The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$
The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
a\...
CommunityBot
- 1
7
votes
1
answer
432
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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}+...
CommunityBot
- 1
3
votes
1
answer
398
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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 ...
- 43.7k
18
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3
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6k
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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 ...
- 121
5
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1
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455
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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
3
votes
3
answers
670
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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.
...
- 1,075
4
votes
1
answer
928
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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
1
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1
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453
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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 ...
- 3,908
4
votes
0
answers
369
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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
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2
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876
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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 ...
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