All Questions
Tagged with nt.number-theory prime-numbers
518 questions with no upvoted or accepted answers
2
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0
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66
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How to check that a number probably/likely has a divisor having a specific bit length/in range?
Given a randomly generated $\alpha\in\Bbb N$ where $\alpha$ is large thus hard to factor (no small prime composites). How to check that a divisor $F\in\Bbb N$ with a specific bitlengh $n\in\Bbb N∧n<...
- 87
2
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0
answers
121
views
How to know if a random natural number is a probable semiprime?
Let that $n\in\Bbb N$ generated from a hash function where $n$ is long enough to be hard to factor in the gnfs algorithm. How to check if $n$ is probably a semi‑prime in a faster way than factoring it ...
- 87
2
votes
0
answers
147
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Prime splitting in the division field of an elliptic curve
Let $E/\mathbb{Q}$ be an elliptic curve with good reduction at two distinct primes $p, \ell$. Suppose the mod $\ell$ Galois representation associated to $E$ is surjective. Let $K=\mathbb{Q}(E[\ell])$ ...
- 1,422
2
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124
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On the elliptic curve $X^3+6d^2X-7d^3 = Y^2$ and the ellipse $p^2+3q^2-d = 0$?
From the ellipse $p^2+3q^2 - d = 0$ we can find a solution to the equation,
$$a^3+b^3+c^3 = (c+m)^3$$
if we solve the elliptic curve,
$$E:=X^3+6d^2X-7d^3 = Y^2$$
More details can be found in this MSE ...
- 12.6k
2
votes
0
answers
131
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Limit of scaled infinite sum with Dirichlet characters modulo 4: is it zero?
I am trying to get an asymptotic formula such as
$$ L_4(s, n) \sim L_4(s) + \rho_n(s)\Lambda_n + \frac{\alpha(s)}{\sqrt{n}} + \frac{\beta(s)}{\sqrt{n\log n}}+\cdots$$
where $L_4(s, n)$ is the first $n$...
- 3,098
2
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0
answers
120
views
On the integer of the form p^a q^b closest to a given integer N
If we give ourselves a number having only one prime factor $p$ and a given natural integer $N$, we know how to give the integer of the form $p^k$ closest (and less than) to this integer $N$ it's ...
- 69
2
votes
0
answers
199
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Not a twin prime pair test using $\gcd$ only
Let $m$ be an odd positive integer such that $m=2k+1$, $k\in\mathbb{N}$.
Let $v$ be a vector of $n$ positive integers. Let $v(i)$ be the $i$-th element of the vector. Then we start with $v(i)=m(i+1)-2$...
- 4,938
2
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0
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103
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On equidistribution of primes in positive characteristic
In S. Lang's book "Algebraic Number Theory" (1986), page 317, Theorem 6 states essentially that given $P$ a set of primes, let $\tau:P\longrightarrow J$ be the typical idèle map taking ...
- 675
2
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70
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Twin prime distribution centering twice a semiprime
What is the conjectured distributional behavior of semiprimes $pq$ ($p$ and $q$ are primes) having the property $2pq+1$ and $2pq-1$ are primes?
- 13.9k
2
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238
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Possible regularisation for sum of function of primes
Consider the following sum of function of primes:
$$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$
Here $p$ runs through all primes and $e$ is Euler's constant.
We can see that the sum ...
- 149
2
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0
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352
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An approximation for the prime counting function
NOTE: I've edited the question one last time, to be much simpler, in the hopes of getting more responses.
SETUP: Let $p_n$ denote the $n$th prime, let $p_x = p_{\lceil x \rceil}$ for all $x > 0$, ...
- 4,985
2
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0
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93
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Primes of the form $a+b^k$ for $k=(a \bmod 2),\ldots,n$?
Procrastinal problem: Given $n$, one can ask for integers $a,b>1$ of different parities
such that $a+b^k$ is prime for $k=(a\bmod 2),\ldots,n$.
A few examples are:
$2+4995825^k$ is prime for $k=0,\...
- 17.9k
2
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0
answers
198
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Generalized primality test for Mersenne and Wagstaff numbers
Inspired by the paper "Chebyshev polynomials and higher order Lucas Lehmer algorithm" by Kok Seng Chu, I think a made a generalized primality test for Mersenne and Wagstaff numbers.
Here is ...
- 119
2
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244
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Lower bounding the number of Galois radii of an integer
Recall that I call $r>0$ a Galois radius of an integer $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ primes and positive $a$ and $b$ and a primality radius of $n$ if $a=b=1$.
Does it suffice to ...
- 6,984
2
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0
answers
263
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Selberg's 1943 result on primes in short intervals and primality radius
This preprint: https://arxiv.org/abs/2207.05038 states in the last paragraph of the first page that a result of Selberg (1943) implies that under RH, almost all intervals of the form $(x,x+\left(\log ...
- 6,984
2
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0
answers
158
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Quadratic patterns in summands of Goldbach's conjecture
Let $n $ be even and define
$$ Q(n)=\sum_{\substack{ p,q \ \textrm{ primes} \\p+q=n }}\left(\frac{p}{q} \right),$$ where $\left(\frac{p}{q} \right)$ is the quadratic Legendre symbol.
Has this sum been ...
- 3,062
2
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125
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Conditional stronger bounds on Linnik theorem with prime power modulus
This post is related to questions asked here and here. However, I include the relevant background on least prime in arithmetic progressions presented here for benefit of the reader.
By Linnik's ...
- 1,042
2
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300
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How soon can we represent a number as the sum of two primes?
Posting in MO since it was unanswered in MSE.
Goldbach's conjecture says that every even number can be represented as the sum of two primes. But how soon can we find such a representation. Taking $20 =...
- 5,470
2
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0
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98
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Primes as expected values?
This is a follow-up question, which is related to the answer of this quesiton: Is there a connection of prime numbers and extreme value theory?
I will duplicate the answer here, so this question is ...
- 3,064
2
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answers
144
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A conjecture about prime test
Conjecture If $\varphi(m)<\varphi(n)$ for all $m<n$,then $n$ is a prime number.
I tried to find a counterexample when $n=pq$ ($p,q$ are prime), then we have to find a prime between $(p-1)(q-1)$ ...
- 141
2
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0
answers
115
views
Error term related to partial twin prime constant
The twin prime constant is defined as
$$
\Pi_2 = \prod_{\substack{p\in\mathcal{P}\\p>2}}\frac{p-2}{p-1}\left(1-\frac{1}{p}\right)^{-1},
$$
where $\mathcal{P}$ is the set of primes. I'm interested ...
- 965
2
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0
answers
136
views
Average length of consecutive integers which have an increasing number of divisors
Consider the nine consecutive natural numbers starting from $1584614377$.
...
- 5,470
2
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0
answers
269
views
A relation of the prime counting function $\pi$ to counting the ordered ways of a number $n$ as a sum of two primes and two questions?
The definitions are from these two questions:
https://math.stackexchange.com/questions/3164216/a-series-related-to-prime-numbers
https://math.stackexchange.com/questions/4349186/trying-to-understand-...
- 3,064
2
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0
answers
87
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The covariance of certain random variable
We define two random variables $X_n,Y_n $ on the sample space $\{1,2,3,\cdots,n\}$ with counting measure. We denote by $C_n$ the covariance of theses two random variables: $C_n=Cov(X_n,Y_n)$.
...
- 356
2
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108
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How to compute/estimate the least $k$ such that there exist $n$ consecutive integers each having a prime factor $\le k$?
Let $a_n$ be the least integer $k$ such that there exist $n$ consecutive integers each with a prime factor $\le k$. For example, $a_{13} \le 11$ because the 13 consecutive integers $114,115,\ldots,126$...
- 523
2
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0
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216
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Two conjectures about generalised A329369
Let $m \geqslant 2$ be a fixed integer.
Let
$$\operatorname{wt}(n,m)=\operatorname{wt}\left(\left\lfloor\frac{n}{m}\right\rfloor,m\right)+n\bmod m, \operatorname{wt}(0,m)=0$$
Then we have an integer ...
- 4,938
2
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0
answers
243
views
Primes in arithmetic progression
We call a prime $p$ "good" if there is $0<k<\log p$ with $2kp+1$ prime. What is the asymptotic density of good primes?
2
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157
views
How could this difference in series of power of zeros associated to counting integers and counting primes be explained?
Introduction:
In this 1992 paper, J.B. Keiper (an amazing person, who tragically died way too young), derives several power series expansions of the Riemann $\xi$-function that involve infinite sums ...
- 4,169
2
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0
answers
205
views
Sum of all primes below $n$ without listing all primes below $n$
Asymptotically there is around $\frac{n}{\ln n}$ primes below a given integer $n$. Thus $\frac{n}{\ln n}$ is a lower bound for the time complexity of any algorithm that at some point finds each prime ...
- 21
2
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0
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313
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Proving that the Riemann zeta function is zero free on Re=1 using the prime number theorem
Is $\frac{-\zeta'(s)}{\zeta(s)}+\frac{-s}{s-1}$ an analytic continuation, holomorphic for $Re\ s > 0,\ s\neq 1$, of $f(s)=s\int_{1}^{\infty}\frac{\psi(x)-x}{x^{s+1}}\mathrm{d}x$?
If so: Let $s_{0}$ ...
- 129
2
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94
views
How far ahead do we have to look after $p_n$ to be sure we find another prime $q$ such that $(p_n+q)/2$ is also prime?
If Goldbach's conjecture is true, given a prime $p_n$ is surely possible to find another prime $q$ such that $\frac{p_n+q}2$ is also prime. But I ask: how far ahead do we have to look after $p_n$ to ...
- 1,008
2
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0
answers
219
views
Is this limit zero?
Define $e(\theta)=e^{2\pi i\theta}, \theta\in [0,1]$, $P_n=\{p_1,p_2,...,p_n\}$ are the first $n$ primes, $\|f\|_1=\int_{[0,1]}|f(\theta)|d\theta$.
Problem 1.
is it true for all fixed $m\in \mathbb{N^...
- 543
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answers
114
views
A conjectured upper bound for the mean value of prime divisors inside prime gaps
In 1969 C.A. Grimm stated this interesting conjecture: the prime gap $\,G_n=\{x\in N:p_n\lt x\lt p_{n+1}\}\,$ contains at least $\,\#G_n=(p_{n+1}-p_n)-1=g_n-1\,$ distinct prime divisors, that is if $\,...
- 1,008
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422
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Sequences with high densities of primes: how to boost them to get even more and larger primes
I propose a methodology to help find large prime numbers with a much higher probability than picking up random numbers and testing them for primality. This would help speed up prime number generators ...
- 3,098
2
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326
views
Why can one compute the sum of divisors of $n$ without factoring $n$?
Question links to paper
which states:
$$
\sigma(n)= \frac{6}{n^2(n-1)}\sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\sigma(n-k) \qquad (1)
$$
where $\sigma(n)$ is the sum of divisors of $n$.
Another similar ...
- 25.4k
2
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96
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if such counter example exists for Lehmer's totient problem could we prove that there are infinity of them or just finitely?
I asked this question one month Ago in MSE but no answer for existence of argument which show if such counter example exists we would have infinity of them or just finitely many examples
Lehmer's ...
2
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answers
313
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On the Chowla and twin prime conjectures
I'm reading https://terrytao.wordpress.com/tag/chowlas-conjecture/ and at some point it is mentioned that, the twin prime conjecture is a variant of Chowla's conjecture that $\sum_{n\leq x} \lambda(n)\...
- 1,019
2
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0
answers
289
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A nice pattern about Goldbach conjecture in French Wikipedia
In the following link: https://fr.m.wikipedia.org/wiki/Conjecture_de_Goldbach, one can see a nice pattern of pink and blue lines coming from each prime number, the intersection points thereof are ...
- 6,984
2
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287
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Best known primality test for the whole intervals of integers up to $10^{20}$ — like the sieve of Eratosthenes
What are the best known primality test(s) for the whole intervals of integers up to $N=10^{20}$ ? "Best" means "have minimal amortized time per tested integer".
That is, the ...
- 121
2
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0
answers
111
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Prime gap transform
Let $n$ be a large enough composite integer, and consider an arithmetic function $f$ that maps $n$ to the sum of prime gaps making a closed interval $J_{f}(n)$ containing $n$ whose extremities are ...
- 6,984
2
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0
answers
203
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Asymptotics on a double sum over primes
I am attemping to find asymptotics of
$$\sum_{p \leq n}\ln p \left( \sum_{k=1}^\infty \left(\left\{\frac{n}{p^k} \right\} - \left\{\frac{n-1}{(p-1)p^k} \right\} \right) - \left\{\frac{n-1}{p-1} \...
- 1,549
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0
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151
views
Is this conjecture equivalent to Polignac's conjecture?
Under Goldbach's conjecture denote by $r_{0}(n)$ for $n$ a large enough composite integer the quantity $\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$, by $k_{0}(n)$ the quantity $\pi(n+r_{0}(n))-\pi(n-...
- 6,984
2
votes
0
answers
198
views
Reciprocity theorem with $n \ge 5$
If $p \equiv 1 \pmod n$, what additional conditions are needed to ensure that
$$2^{(p-1)/n} \equiv 1 \pmod p?$$
I know:
For $n=3$ (cubic reciprocity) the form is $p=x^2+27y^2$.
For $n=4$ (...
- 131
2
votes
0
answers
97
views
Improved upper bound for second moment of reduced residues modulo $q$?
The question asked here is a follow up to this question, which was answered by user GH from MO. [EDIT: The question is edited for clarification after receiving a comment on the original posting.]
As ...
- 965
2
votes
0
answers
84
views
The number of admissible tuples with last element equal to $h_{k-1}$?
Let $k \geq 2$ and $(h_1, h_2,\cdots,h_{k-1}) \in \mathbb{N}^{k-1}$.
Consider the $k$-tuple : $\mathcal{H}_k=(0,h_1,\cdots,h_{k-1})$ with $0<h_1<\cdots<h_{k-1}$.
The $k$-tuple $\mathcal{H}...
- 521
2
votes
0
answers
537
views
Explicit formula for $n$th prime in terms of Riemann zeros:
We all know there exists an explicit Formula for prime counting function in terms of Riemann zeros.
I'm wondering if similar formula exists for $n$th prime in terms of Riemann zeros?
Or any other ...
- 375
2
votes
0
answers
172
views
Funny questions about Moebius Function
I need to firstly claim that my research is not about number theory, however, I am pretty interested in it, especially funny questions in number theory, e.g. Kollatz Conjecture. Three years ago, I ...
- 41
2
votes
0
answers
156
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Questions about a certain sequence of naturals generated by primorials
I'm working on the following sequence of naturals (which is NOT listed in OEIS)
$$3,5,11,17,23,29,59,89,119,149,179,209,419,629,839,1049,1259,1469,1679,...$$
whose elements are generated this way
$$3=(...
- 1,008
2
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0
answers
195
views
Error bounds for $\pi(x)-Li(x)$ and convergence of the associated Dirichlet integral
Following On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$, define
$$I_{s}=\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x,$$ where $\pi$ denotes the prime counting function ...
- 1,019
2
votes
0
answers
131
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How many divisors of $\phi(m)$ do not divide $m-1$?
Lehmer's totient problem asks if there exists a composite number $m$ such that $\phi(m)$ divides $m-1$. Lower bounds on $m$ has been established but we do not know if a solution exists. Clearly, if we ...
- 5,470