All Questions
Tagged with nt.number-theory prime-numbers
315 questions
11
votes
2
answers
615
views
Jacobi symbols for two-square sums of primes
Given a prime $p\equiv 1\pmod 4$, Fermat's two-squares theorem discovered by Girard
states that there exists two integers $A,B$ such that
$p=A^2+B^2$.
For all primes up to $10^7$ the integers $A$ and $...
- 17.9k
11
votes
0
answers
1k
views
Are the twin primes the only positive double zeros of this real function?
Agno's answer
was extremely helpful.
For $x \in \mathbb{R}, x \ge 1$ define
$$ f(x) = \sin\frac{\pi(\Gamma(x)+1)}{\lfloor x \rfloor}$$
By Wilson's theorem the positive integer zeros of $f(x)$ are ...
- 25.4k
11
votes
0
answers
565
views
Polynomial mapping primes to primes
Consider a non constant polynomial $P\in\mathbb{Z}[X]$ sending prime numbers to prime numbers. I encountered on the web two different proofs that $P$ is the identity polynomial, one on mathoverflow ...
- 3,425
11
votes
1
answer
637
views
Primes such that a given number has very small order
The following came up in (a previous version of) this answer.
Question. Let $a > 1$ be a positive integer, and $f \in \mathbf Z[x]$ a polynomial with positive leading term. Does there always exist ...
- 28.7k
11
votes
2
answers
1k
views
Chebyshev's approach to the distribution of primes
This is motivated by a recent question by Wadim.
The negative answer should be known, since t is very natural, in this case I would be happy to see any reference.
May Pafnuty Lvovich Chebyshev's ...
- 109k
10
votes
2
answers
2k
views
Abel summation of the alternating series of primes?
Consider the ordinary generating function of the sequence of primes ($2+3x+5x^2+7x^3 + ...$); by the ratio test and the prime number theorem, its radius of convergence is $1$. Thus, we might well ask ...
- 5,827
10
votes
1
answer
2k
views
What keeps asymptotic Goldbach's conjecture out of reach of current technology?
Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...
- 6,984
10
votes
1
answer
2k
views
Convergence of the series $\sum_p p^{-s}$ ($p$ prime and $s>1$)
I know that $\sum_p p^{-s}$, $s>1$, converges. Now, I define $J(s) = \sum_p p^{-s}$. Are there any "well known" values for $J(2)$, $J(3)$, $J(4)$, etc? We all know that $\zeta(2)= \frac{\pi^2}{6}$, ...
- 101
10
votes
2
answers
1k
views
Is $n = p-q$ equivalent to Goldbach's Conjecture?
One open conjecture is that every even integer greater than two is the difference of two primes. (Some superficial discussion here.)
Goldbach's conjecture states that every even integer greater than ...
- 201
10
votes
2
answers
1k
views
What is the lower bound for highly composite numbers?
if $x=d(n)$ is the number of divisors of $n$, what is the tightest lower-bound for $n$ only given $x$?
http://en.wikipedia.org/wiki/Highly_composite_number
- 105
10
votes
2
answers
1k
views
Is a Galois extension of the rationals determined by its set of completely split primes?
apologies if this is a naive question. Consider two Galois extensions, K and L, of the rational numbers. For each extension, consider the set of rational primes that split completely in the extensions,...
- 595
10
votes
1
answer
694
views
Prime numbers from permutation
Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n)$ fixed and reverse-cyclically permuting every $n$ consecutive terms thereafter; apply $P(2)$ to $1,2,3,...$ to get $...
- 4,938
10
votes
0
answers
205
views
Does the diophantine equation $\,\prod_{k=1}^n(p_k^{x_k}-1)=y^2\,$ have always at least a solution for $\,n\gt2\,$?
P.G.Walsh proved in this paper that the diophantine equation $\,(2^{x_1}-1)(3^{x_2}-1)=y^2\,$ has no solution in positive integers $\,x_1$, $\,x_2\,$ and $\,y$.
If we generalize the previous equation ...
- 1,008
10
votes
4
answers
1k
views
The smallest solution to $2^{2k}-1=\text{powerful}$
Integer is powerful if all the exponents in its factorization are at least $2$.
Every powerful integer can be written in the form $a^2 b^3$.
For odd $k$, define $F(k)=2^{2k}-1=(2^k-1)(2^k+1)$.
This ...
- 25.4k
9
votes
2
answers
1k
views
Random pseudoprimes vs. primes
(Edit. What I called "pseudoprimes" are known as "Cramér random primes" in the literature,
of which I was unaware.)
Say that a set $S$ of natural numbers is a set of pseudoprimes if they
are (a) ...
- 151k
9
votes
0
answers
324
views
Semi-primes represented by quadratic polynomials
According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(...
- 195
9
votes
1
answer
698
views
Hensel's lemma, Bezout's identity, and the integers
Factorization in the ring $\mathbb{Z}[x]/(x^2+1)\mathbb{Z}[x]\cong \mathbb{Z}[i]$ is well known. For instance, $5$ and $13$ (and any prime $\equiv 1\pmod{4}$) are no longer prime.
The factorization ...
- 18.7k
9
votes
3
answers
2k
views
May $p^3$ divide $(a+b)^p-a^p-b^p$?
Do there exist positive integers $a,b$ and a prime $p>\max(a,b)$ such that $p^3$ divides $(a+b)^p-a^p-b^p$?
The reader of Kvant magazine A. T. Kurgansky asked to prove that such $a,b,p$ do not ...
- 109k
9
votes
5
answers
2k
views
Optical methods for number theory?
I found a paper: 'A New Method of Finding the Distribution of Prime Number', saying
We stack discs and annuluses with certain rules then turn on the light to illuminate. The projection of ...
9
votes
3
answers
680
views
Quadratic residues and nonresidues of arbitrary patterns
Let $p_1, p_2, \dotsc, p_n$ be distinct primes, and let $\epsilon_1, \epsilon_2, \dotsc, \epsilon_n$ be an arbitrary sequence of $1$ and $-1$.
There is an integer $a$ such that $\left( \frac{a}{p_1} \...
- 483
9
votes
1
answer
418
views
Conjectured primality test for specific class of $N=4kp^n+1$
Can you provide a proof or counterexample for the following claim?
Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ .
Let $N= 4kp^{n}+1 $ where $k$ is a positive natural number , $...
- 2,661
8
votes
1
answer
811
views
Primes of the form $x^2 + y^2 + 1$
There are infinitely many primes of the form $x^2+y^2+1$, as proved by Bredihin. Motohashi improved the result by showing that there were $\gg x/\log^2 x$ such primes up to $x$. But we expect $\Theta(...
- 9,114
8
votes
1
answer
765
views
An alternative to continued fraction and applications
This post is inspired by the Numberphile video 2.920050977316, advertising the paper A Prime-Representing Constant by Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, ...
8
votes
1
answer
855
views
Is it possible to sum the divergent series with prime coefficients?
It is known that the series $$ P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime} $$ cannot be summed by means of (prime) zeta function regularization. (The result was ...
- 4,829
8
votes
1
answer
1k
views
Would Elliott-Halberstam conjecture follow from GRH?
The Wikipedia article about Elliott-Halberstam (EH for short) conjecture says that the so-called Bombieri-Vinogradov theorem, which is a weaker form of EH conjecture, is in some sense an averaged form ...
- 6,984
8
votes
1
answer
2k
views
Dirichlet's theorem for number fields
I'd like to see a formulation of Dirichlet's theorem for number fields, i.e. some analogue of the assertion:
The number of primes less than $N$ congruent to $a \pmod{m}$ where $(a,m)=1$ is
$\frac{...
- 543
8
votes
1
answer
937
views
On the connection between sums of prime numbers and distribution of prime numbers
As an amateur mathematician, I have always been fascinated by the magic of prime numbers, and their apparently random distribution. I was utterly amazed when I found the following connection between ...
- 189
8
votes
1
answer
605
views
lower and upper bound for $\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k$?
Are there known any lower and upper bounds for
$$
\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k,
$$
where $\Omega(n)$ is the number of prime factors counting multiplicities of $n$?
Or at least is it known ...
- 841
8
votes
4
answers
1k
views
Prime numbers $p$ not of the form $ab + bc + ac$ $(0 < a < b < c )$ (and related questions)
If we ask which natural numbers n are not expressible as $n = ab + bc + ca$ ($0 < a < b < c$) then this is a well known open problem. Numbers not expressible in such form are called Euler'...
- 3,463
8
votes
1
answer
570
views
Asymptotic limit of truncated Legendre sieve
Consider the truncated sum
$$
S(x):=\sum_{\substack{{d\mid P(\sqrt{x})}\\{d\leq x}}}\mu(d)/d,
$$
where $P(z)$ is the product of all primes less than or equal to $z$, and $\mu(d)$ is the Möbius ...
- 965
8
votes
1
answer
838
views
Density of prime pairs whose gap is less than the average gap
By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap ...
- 5,470
8
votes
2
answers
814
views
Estimates about prime numbers: a lemma in Bourgain's article
For $n\in \mathbb{N}$ with prime decomposition $n=p_1^{r_1}\cdots p_k^{r_k},p_i\neq p_j$, let $A=\{p_1,\cdots,p_k\}$; then the following holds:
\begin{equation}
|\{q\in \mathbb{N},q<Q: \text{all ...
- 351
7
votes
3
answers
1k
views
Values where infinite products of primes and composites are equal
Highly grateful for your help/steers on the following question (at the end):
Take the infinite product:
$$\displaystyle T(s) = \prod _{n=2}^{\infty } \left( \dfrac{{n}^{s}} {{n}^{s}-1}\right)$$
for ...
- 4,169
7
votes
2
answers
513
views
A non-standard ergodic limit
Suppose $T$ is an ergodic measure-preserving transformation on a measure space $(X,\Sigma,\mu)$, and $f\in L^1(\mu)$. Does the limit
$\lim_{X\to\infty} \pi(X)^{-1}\sum_{p\leq X} f(T^{p}x)$
exist ...
- 13.1k
7
votes
0
answers
179
views
When does the function $F(x)=\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$ reach $F(x) > 8$?
We know from Ramanujan and Riemann that,
$$\pi(x) = \operatorname{li}(x) -\tfrac12\operatorname{li}(x^{1/2})-\tfrac13\operatorname{li}(x^{1/3})-\tfrac15\operatorname{li}(x^{1/5}) +\dots$$
with prime ...
- 12.6k
7
votes
1
answer
462
views
Primality test for $N=2^a3^b+1$
Can you prove or disprove the following claim:
Let $N=2^a3^b+1$ , $a>0 , b>0$ . If there exists an integer $c$ such that $$c^{(N-1)/3}-c^{(N-1)/6} \equiv -1 \pmod{N}$$ then $N$ is a prime.
You ...
- 2,661
7
votes
2
answers
932
views
A stronger form of the Dirichlet Theorem on prime numbers in arithmetic sequences
Question 1. Let $a,b>1$ be two natural numbers. Is there a prime number $p\in 1+b\mathbb N$ such that $a+p\mathbb Z$ is a generator of the multiplicative group of the field $\mathbb Z/p\mathbb Z$?
...
- 41.9k
7
votes
2
answers
1k
views
Lower bound of the number of relatively primes(each-other) in an interval
I am trying to find lower and upper bounds for the number of integers that are coprime in pairs in an interval of length n.
What are the best bounds that we have?
Is that true that in any interval ...
7
votes
2
answers
834
views
Smoothness in Mersenne numbers?
The $n$-th Mersenne number $M_n$ is defined as
$$M_n=2^n-1$$
A great deal of research focuses on Mersenne primes. What is known in the opposite direction about Mersenne numbers with only small ...
- 3,979
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
7
votes
4
answers
735
views
Sieve of Erathostenes: removing consecutive items
This question comes from https://stackoverflow.com/questions/13747873/why-does-this-prime-function-work, where somebody wrote a standard Sieve of Erathostenes algorithm --- with a bug. However, the ...
- 73
7
votes
2
answers
2k
views
Legendre's Constant
In a couple of web pages, I see that Legendre's constant is defined to be $\lim_{n \to \infty} (\pi(n) - (n/\log(n)))$ (for example, here and here).
Actually the first uses $\lim_{n \to \infty} (\log(...
- 595
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
7
votes
0
answers
843
views
Permutations of the set $\{1,2,...,n\}$ and prime numbers
Here is the version of this question that I posted on math.stackexchange a few days ago and I did not receive an answer that settles my question so I thought that maybe on this site I could get a ...
- 191
6
votes
2
answers
319
views
Evolution of partial sum of a sequence of induced Dirichlet characters
Let's consider the Dirichlet Character $\chi_3(n)$ modulo 3 given by $\chi_3(1)=1$, $\chi_3(2)=-1$ and $\chi_3(3)=\chi_3(0)=0$.
Lets consider the sequence of induced characters $\chi^{P_N} $ obtained ...
- 1,199
6
votes
4
answers
845
views
Uniform upper bound for the sum over primes $\sum_{p \leq x} p^{-1+\varepsilon}$
I am reading the article D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprime, Math. Comp. (1991) (click).
In equation (27) the authors, apparently, used the following ...
user40023
6
votes
0
answers
448
views
Are there always at least *five* divisions?
@JosephO'Rourke asked a question about a Collatz like function related to primes:
$f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is prime} \\
\lfloor n/2 \rfloor & \text{if} \;n \;\text{...
- 1,375
6
votes
1
answer
443
views
Do $2^{n-1}\equiv1\pmod n$ and $(n-1)/2$ prime imply $n$ prime?
Do $2^{n-1}\equiv1\pmod n$ and $(n-1)/2$ prime imply $n$ prime?
Equivalently: Does $n$ being a Fermat pseudoprimes to base 2 (OEIS A001567) imply that $(n-1)/2$ is composite? That holds for all $n<...
- 340
6
votes
1
answer
392
views
Arithmetic properties of positively reduced $2\times 2$-matrices
Call a $2\times 2$ matrix with coefficients in $\{0,1,2,3,\ldots\}$
positively reduced if any row or column reduction (given by replacing a row/column by itself minus the other row/column) produces
at ...
- 17.9k
6
votes
1
answer
672
views
Is there a 2-power-twinless prime?
Call two primes 2-power-twins if their difference is (can you guess?) a power of 2.
For example, 11 and 19 are 2-power-twins.
Is there a 2-power-twinless prime?
I would imagine that this is doable ...
- 19k