All Questions
Tagged with nt.number-theory prime-numbers
518 questions with no upvoted or accepted answers
3
votes
0
answers
85
views
Iteration of a primeness-measuring function
Question
For $n \in \mathbb{N}$ let $\delta(n)$ denote the cardinality of the set $$\left\{(a,b) \in \mathbb{N}^2 \::\: 1 < a < n,\ 1 < b < n,\ n|ab\!\: \right\}.$$ Let $D(n)$ denote ...
- 756
3
votes
0
answers
125
views
On the set $\{n>0:\ n\ \text{is a quadratic nonresidue modulo the}\ n\text{th prime}\}$
Let $S$ denote the set of positive integers $n$ with $n$ a quadratic nonresidue modulo the $n$th prime $p_n$. The first 20 elements of $S$ are
$$2,\, 3,\, 6,\, 7,\, 8,\, 10,\, 11,\, 13,\, 15,\, 18,\, ...
- 15.6k
3
votes
0
answers
194
views
Conjectures for primes $p\equiv1\pmod3$
Let $p$ be a prime with $p\equiv1\pmod3$. It is well known that we can write $p$ uniquely as $a_p^2+a_pb_p+b_p^2$ with $a_p,b_p\in\mathbb Z$ and $a_p>b_p>0$.
Note that $a_b\not \equiv b_p\pmod3$....
- 15.6k
3
votes
0
answers
87
views
Are there infinitely many primes $p$, positive integers $ k, n $ such that $1 \le n < p$ and $p^k > n.rad(p^{k+1}−n)$?
Among $168$ prime numbers in range $1$ to $10^3$, there are $84$ prime numbers $n$ such that: $p^k> n.rad(p^{k+1}−n)$ where $1 \le n<p$ and $k=2,3,4$. There are also $84$ prime numbers $n$ such ...
- 1,776
3
votes
0
answers
673
views
Prime numbers and sieving with $2,3,\cdots,q(x)= (1+o(1)) \log(x)$
Let $x \in \mathbb{R}_{+}$.
For $q \in \mathbb{P}$, let : $\mathcal{B}_q = \{b \in \mathbb{N}^{*} \, | \, \gcd(b, \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p prime}}} {\normalsize p}})=...
- 521
3
votes
0
answers
240
views
Complexity of representations of sets using elementary functions
Fermat conjectured that $2^{2^n}+1$ is prime for every $n \in \mathbb{N}.$ Before even knowing about Euler's counterexample (that $2^{32}+1 = 641 \cdot 6700417$), you could possibly say that Fermat ...
- 745
3
votes
0
answers
183
views
From Firoozbakht's conjecture to set interesting conjectures for sequences or series of primes
In this post we denote the $k-th$ prime number as $p_k$. I present two conjectures, the first about the asymptotic behaviour of a product and the other about an alternating series. I don't know if ...
3
votes
0
answers
299
views
An attempt to get a variant of Agoh–Giuga conjecture
The idea of this post is an attempt to explore a variant of the so-called Agoh–Giuga conjecture. In past days, and today, I tried to think about variants of this conjecture exploring congruences about ...
3
votes
0
answers
280
views
Magnitude and distribution of largest prime factor?
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors.
What is magnitude and distribution of largest prime factor of typical magnitude $n$ natural number?
What is ...
- 13.9k
3
votes
0
answers
266
views
Conjectured primality test for numbers of the form $N=4 \cdot 3^n-1$
This is a repost of this question.
Can you provide proof or counterexample for the claim given below?
Inspired by Lucas-Lehmer primality test I have formulated the following claim:
Let $P_m(x)=2^{-m}\...
- 2,661
3
votes
0
answers
171
views
Estimating integral of product of terms $\cos(t\log p)$
I would like to prove the following proposition from A. Harper's paper "Sharp conditional upper bound for moments of the Riemann Zeta Function"
Proposition.
Let $T$ be large and let $n=p_1^{\...
- 199
3
votes
0
answers
203
views
Longest known polynomial progression of distinct primes
Is Euler’s quadratic progression of forty distinct primes (the values of $n^2-n+41$ for $n$ between 1 and 40) still the longest known sequence of this kind?
I’d also be curious to know the longest ...
- 19.7k
3
votes
0
answers
121
views
Does $(p-1)^2$ divide $\det[(\frac{i^2+cij+dj^2}p)]_{0\le i,j\le p-1}$ when $(\frac dp)=-1$?
Let $p$ be an odd prime. As in my paper, for $c,d\in\mathbb Z$ let us define
$$[c,d]_p:=\det\left[\left(\frac{i^2+cij+dj^2}p\right)\right]_{0\le i,j\le p-1},$$
where $(\frac{\cdot}p)$ is the Legendre ...
- 15.6k
3
votes
0
answers
131
views
Chen primes and permutations
In 1973 the Chinese mathematician J.-R. Chen proved that there are infinitely many primes $p$ such that $p+2$ is a product of at most two primes. Nowadays such primes $p$ are called Chen primes.
For $...
- 15.6k
3
votes
0
answers
293
views
Primes arising from permutations (II)
In Question 315259 (cf. Primes arising from permutations) I asked a question on primes arising from permutations which looks quite challenging.
Here I pose a new question in this direction which does ...
- 15.6k
3
votes
0
answers
206
views
Cancellation in this exponential sum?
I would like to know whether it is possible to obtain cancellation in the sum
$$\sum_{p \leq X} e^{{2\pi iX}/{p}}$$
where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number.
- 11.3k
3
votes
0
answers
154
views
Is there a name for sequences of integers reduced to their lowest prime divisors?
When trying to obtain the value of Jacobsthal's function for some $n$; to find the largest sequence of consecutive numbers that are all coprime to $n$, one approach (and the only direct approach that ...
- 161
3
votes
0
answers
265
views
Prove A Skipping Prime Conjecture For Rio?
I am writing a paper to accompany a Short Communication I plan to give in Rio this August. The paper regards work on jumping primes, a project on which Jose Brox has been working with me. I was going ...
- 13k
3
votes
0
answers
259
views
Deduction of the classical Halasz's inequality from the new form
In the new proof of Halasz's Theorem, the authors give a statement in terms of a quantity $L(x)$ defined by $$L(x)^2=\sum_{|N|\leq\log^2x+1}\frac1{N^2+1}\max_{|t-N|\leq1/2}|F_x(1+it)|^2,$$
where $F_x(...
- 63
3
votes
0
answers
138
views
Is the conjunction of Goldbach and NFPR conjecture actually equivalent to Hardy-Littlewood k-tuple conjecture?
In this previous question of mine
I introduce under Goldbach's conjecture the notation $ r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $ as well as the related so-called NFPR conjecture ...
- 6,984
3
votes
0
answers
319
views
Isometry group of an integer
This is a cross post from MSE, as it seems the partial answer I got then was deleted, so I ask it again here.
Let $\prod_{i\in I}p_{i}^{a_{i}}$ be the prime factorization of a positive integer $n$ ...
- 6,984
3
votes
0
answers
153
views
Short intervals containing a prescribed number of primes
In arXiv:1802.10327, the following conjecture appears:
$$\vert\{n\leq x,\vert [ n,n+\lambda\log n]\cap\mathbb{P}\vert=m\}\vert\sim\frac{\lambda^{m}e^{-\lambda}}{m ! }\cdot x$$
Differentiating the ...
- 6,984
3
votes
0
answers
286
views
What is the value of this simple game with primes?
Consider the following game. Alice selects an integer $n$ from $[1,b]$, while Bob selects an integer $m$ from $(a,b]$ (for concreteness, you may choose $a=10^{10}$ and $b=10^{1000}$). Alice wins if $m-...
- 781
3
votes
0
answers
131
views
Improving prime number generation probability?
Deterministic generation of primes in polynomial time is unknown.
Is there a way to probablistically in $O(n^c)$ time bound for some $c>0$ generate polynomially $\Omega(n^c)$ many integers in $[0,...
- 13.9k
3
votes
0
answers
408
views
The second conjecture about the degrees of special polynomials
Define the congruence "modulo m" on exponential Taylor series following the previous post (A conjecture about the degrees of special polynomials)
It has been conjectured, that if we define the ...
- 237
3
votes
0
answers
188
views
Riemann's explicit formula for square-free numbers
We know that for $x$ being a half-integer $$\sum_{n\leq x}\Lambda(n)=x-\sum_{\rho}\frac{x^\rho}{\rho}+O(1).$$
Is there a similar formula for $\sum_{n\leq x}\mu(n)^2$ in the literature? The underlying ...
- 3,062
3
votes
0
answers
334
views
Conditional proof of ternary Goldbach
This is a reference request.
I know that Hardy and Littlewood gave a proof of the ternary Goldbach for sufficiently large odd integers under the assumption of GRH.
Is there a modern account of ...
- 3,062
3
votes
0
answers
111
views
On covering with Idoneal integers
$d\in\Bbb N$ is an idoneal integer if $N\in\Bbb N_{>1}$ can be written uniquely as $N=x^2\pm dy^2$ then $N=2^mp^n$ where $p$ is odd prime and $n\geq0$ and $m\geq0$ holds.
Let the $65$ known ...
user94040
3
votes
0
answers
135
views
The probability of having $\omega(z_1\dotsb z_\kappa)<\kappa$
The problem below is suggested by this one.
Suppose that we are given $2k$ integers $x_1,\dotsc,y_k$, and we want to find an integer $a$ so that $\gcd(a+x_i,a+y_i)>1$ for each $i\in[1,k]$. This ...
- 23k
3
votes
0
answers
139
views
Square integral of finite Euler product
Consider the finite Euler product
$$
P(t) = \prod_{r=1}^R \left(1 + p_r^{i t} \right).
$$
(Here $p_1, p_2, \dots$ are of course the primes.)
Question: What is a good asymptotic upper bound for
$$
\...
- 2,207
3
votes
0
answers
134
views
Distribution of the inbetween prime
Let $\ \mathbb J_n\,:=\,\{1\ \ldots\ n\}\ $ be the initial interval of natural numbers, and
$$2=p_0<p_1<\ldots$$
be the increasing sequence of all primes. Let
$$ \forall_{n=1\ 2\ \ldots}\ \ d_n\...
- 7,500
3
votes
0
answers
320
views
On sets of coprime numbers
We know that from prime number theorem that the number of primes below $n$ and above $\frac n2$ (denoted by $\pi_{n,\frac n2}$ is approximately $$\pi_{n,\frac n2}\approx\frac{n}{2\ln n}.$$
Denote by $...
user76479
3
votes
0
answers
443
views
Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula?
My question is on Mobius inversion formula convergence/properties when used with infinite sums of function.
Lets consider (on $\mathbb{R}^{+}$):
$$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$
We call $...
- 1,199
3
votes
0
answers
321
views
Density of numbers whose prime factors all come from a fixed congruence class
Let $q$ be a positive integer greater than one, and let $a$ be an integer such that $\gcd(a,q) = 1$. Define
$$D(a,q) = \{n \in \mathbb{N} : p | n \Rightarrow p \equiv a \pmod{q} \}.$$
Do we know the ...
- 26.9k
3
votes
0
answers
369
views
Metric on the set of subsets of the rational primes
Note: this is a revision of an earlier post. It was kindly pointed out that my initial proposed metric was in fact not a metric, so this is a revised version.
I was thinking how to say that two sets ...
- 595
3
votes
0
answers
317
views
Prime Hadamard matrices
Assume that $n$ is a sufficiently large number. Is there a Hadamard matrix $H_{4n \times 4n}=(h_{ij})$ with the last row and the last cloumn $J$ (thet is, for every $k$, $h_{k,4n}=1$ and $h_{4n, k}=1$)...
- 321
3
votes
0
answers
477
views
Numbers expressible as sums of prime powers larger than n
Given a fixed $n \in \mathbb{N}$ larger than $1$, let $G(n)$ denote the largest number that is not expressible as a sum of prime powers larger than $n$ (the 'base' prime of the prime power need not be ...
- 93
3
votes
0
answers
297
views
An inequality about Goldbach conjecture
Let $N$ a large natural number, let $\forall n\leq N,\, R_{2}\left(n\right)=\underset{p_{1}+p_{2}=n}{\sum}\log\left(p_{1}\right)\log\left(p_{2}\right)$ and let $S\left(\alpha\right)=\underset{p\leq N}{...
- 319
3
votes
0
answers
251
views
Logarithms of ratios of squarefree numbers
Let $M \geq 1$, and $N=2^M$. Let $a_1, \dots, a_N$ be the set of all the numbers that you get when forming all square-free products of the first $M$ primes.
For example, for $M=2$ and $N=4$ you get $...
- 2,207
3
votes
0
answers
919
views
Linear independence over Q of logarithmic powers of prime numbers
I denote $p_k$ the $k^{th}$ prime number ($p_1=2$, etc...)
Clearly, for any $n\in \mathbb{N}^*$, $(\log p_k)_{1\leq k\leq n}$ is linearly independent over $\mathbb{Q}$.
My question concerns a ...
- 101
3
votes
0
answers
1k
views
Effective upper bound on large prime gaps; or, what is the first prime after a googolplex?
Question
What is the best known effective upper bound on the prime gap following x?
Motivation
Suppose you needed to show a good bound for the gap between a fixed large constant, say $G=10^{10^{100}...
- 9,114
3
votes
0
answers
1k
views
Generating random smooth numbers (or, "What do random smooth numbers look like?")
What is a good method for generating random b-bit, S-smooth numbers?
For S large and b not too large, it may be feasible to generate random numbers and test if they are smooth enough. If S is too ...
- 9,114
2
votes
0
answers
59
views
Wieferich primes and identities for the Euler quotients of $2^n+1$ and $\frac{2^n+1}{3}$
Let $n>1$ be odd integer.
Define the Euler quotient $a(n)=\frac{2^{\varphi(n)}-1 \bmod n^2}{n}$.
Number $n$ with $a(n)=0$ is Wieferich number and if it is prime
it is Wieferich prime.
It is open ...
- 25.4k
2
votes
0
answers
110
views
+50
How to apply Pohlig Hellman using a very limited set of auxiliary inputs in that case?
So I was reading about Talotti, Paier, and Miculan - ECC’s Achilles’ Heel: Unveiling Weak Keys in Standardized Curves. The underlying idea is to lift the discrete logarithm problem to $\mathrm{prime}−...
- 87
2
votes
0
answers
76
views
upper and lower bounds on rowlands sequence
rowlands sequence is defined as follows
\begin{equation}
a_{n}=a_{n-1} + b_{n}
\end{equation}
where $b_{n} = gcd(a_{n-1}, n)$ for $n>h$
it originates from E. Rowlands 2008 paper "A Natural ...
2
votes
0
answers
108
views
Largest prime determinant of a binary matrix
Given an integer $n$, I want to prove the existence of an $n\times n$ binary matrix (with 0,1 entries), whose determinant is a prime number. What is a lower bound on the largest determinant that I ...
- 3,549
2
votes
0
answers
191
views
The exponential sum over primes on average
In https://academic.oup.com/blms/article-abstract/20/2/121/266256?redirectedFrom=fulltext Vaughan shows the following bounds for the $L^1$-mean of the exponential sum over primes $$\sqrt x\ll \int _0^...
- 1,381
2
votes
0
answers
182
views
Integers as polynomials in infinite variables
This question is more of a request for reference or ideas than else. Forgive (or correct) if there are imprecisions or blatant mistakes.
The main idea is that the unique factorization theorem for $\...
- 29
2
votes
0
answers
411
views
On two "versions" of abc conjecture
Let $a,b,c$ be coprime nonzero positive integers such that $a+b=c$. The ABC conjecture states that for any $\varepsilon>0$, we have $$c < C_{\varepsilon}\operatorname{rad}(abc)^{1+\varepsilon}.$$...
- 622
2
votes
0
answers
157
views
Conjecture: $x^4+1$ is never Wieferich prime
Related to this question and Alexander Kalmynin's answer.
For natural $n$ define $J(n)=(2^{n-1}-1) \bmod n^2$
and if $n$ is power of two define $J(2^n)=1$ (this is artificial, just to
avoid triviality ...
- 25.4k