All Questions
Tagged with nt.number-theory prime-numbers
518 questions with no upvoted or accepted answers
7
votes
0
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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
0
answers
546
views
Is every integer greater than 1 the sum of a palindrome and a prime?
Helfgott proved that any odd number greater than 5 is the sum of three primes. Cilleruelo and Luca proved that every positive integer is the sum of three palindromes.
Is every integer greater than 1 ...
- 3,707
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
7
votes
0
answers
267
views
Can primes be (almost) random sequence in von Mises sense?
Random models for primes (such as Cramer's model) have been extensively used for informal justification of various conjectures involving primes. It is crucial to understand in what sense sequence of ...
- 7,182
7
votes
0
answers
860
views
Are there an infinite number of twin semiprimes?
A semiprime is a number that is the
product of two (possibly equal) primes.
Define twin semiprimes (my terminology) as two consecutive numbers both semiprimes.
For example, $(57,58)$ are twin ...
- 151k
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
0
answers
294
views
On the ratio of Gilbreath sequences
Definitions: let $n \in \mathbb{N}_{>0} \cup \{ \infty \}$ and let $E_n$ be the set of sequences $(d_i)_{i=1}^n$ such that $d_1=1$, $d_i$ is an even integer (for $i > 1$) and $0<d_i \le i$. ...
7
votes
0
answers
959
views
Primes for which 2 is a primitive root
I am writing a paper in which I keep referring to primes p for which 2 is a primitive root mod p and so I want to give a name for these primes. Is there a name for these primes in the literature ...
- 575
7
votes
0
answers
492
views
Twin Primes that are Sophie Germain Primes
Suppose $p$ is a prime such that $p + 2$ is also prime, and nothing else is known about $p$.
Is there any reason to think that this affects the probability that $p$ is also a Sophie Germain prime? ...
- 123
7
votes
0
answers
870
views
Generating function for the characteristic function of prime numbers
What do we know about the generating function of $\chi(n)$ (A010051)
$$
f(x) = \sum_{n=0}^\infty \chi(n)x^n = \sum_{p\text{ prime}} x^p
$$
for $\chi(n)$ the characteristic function of the primes:
$$...
- 191
7
votes
0
answers
332
views
$n\varphi(n)\equiv 2\pmod{\sigma(n)}$ as a primality test
It is known from Subbarao, "On two congruences for primality" that $n>22$ is a prime iff $$n\sigma(n)\equiv 2\pmod{\varphi(n)},$$ where $\varphi(n)$ is Euler's function and $\sigma(n)$ is sum of ...
- 12.3k
7
votes
0
answers
452
views
Primes of the form $x^2+ny^2$ such that when you swap x and y you get another prime
Hello, Im looking at primes of the form $x^2+ny^2$ for $n>1$ where we can swap $x$ and $y$ and get another prime, I have found many such pairs for many values of n, and i wanted to know if there ...
- 787
7
votes
0
answers
709
views
"probabilistic" density of primes?
A certain set $\cal P$ of primes is defined by two assumedly independent conditions:
The first condition on a prime $p$ can be characterized in terms of the type of splitting of $p$ in certain Galois ...
- 797
6
votes
0
answers
381
views
A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
- 81
6
votes
0
answers
169
views
$p^2+a^2$ can be a squarefree number with all prime divisors less than $p$?
Let $p$ be a prime $\ge 31$.
Is there an integer $a < p$ such that $p^2 + a^2$ is a squarefree and all of its prime divisors are less than $p$?
For example, for $p=31$, $31^2+5^2 = 986 = 2 \times ...
- 483
6
votes
0
answers
126
views
Equivalence of primes based on the partition of their Pisano periods
The period of Fibonacci numbers modulo $m$ is called Pisano period and its length is denoted as $\pi(m)$. Define the Pisano partition of $m$ as the set partition of the indices $\{0,1,\dotsc,\pi(m)-1\}...
- 34.3k
6
votes
0
answers
230
views
A bias for runs in Legendre symbols?
$\newcommand\Legendre[2]{\genfrac(){}{}{#1}{#2}}$An odd prime $p$ defines the sequence $\Legendre1 p,\Legendre2 p,\dotsc,\Legendre{p-1}p$
of values of the Legendre symbol describing the quadratic ...
- 17.9k
6
votes
0
answers
479
views
Existence of an explosive prime
The motivation to introduce explosive prime is Carmichael's totient conjecture (see why below).
Let $\mathbb{N}_{SF}$ be the set of positive square-free integers. Consider the map $f:\mathbb{N}_{SF} \...
6
votes
0
answers
149
views
Dickson's conjecture for Beatty sequences
A particular case of Dickson's Conjecture states that for $a_1,q_1,a_2,q_2$ with $(a_1,q_1)=(a_2,q_2)=1$, there are infinitely many $n$ for which $q_1 n + a_1$ and $q_2 n+a_2$ are both prime, provided ...
- 1,990
6
votes
0
answers
192
views
A conjecture involving $P_n=\prod_{k=1}^np_k$
For each positive integer $n$ let $P_n=\prod_{k=1}^n p_k$, where $p_k$ is the $k$th prime.
Question. Is my following conjecture true?
Conjecture. For any integer $n>1$, there are $k,m\in\{1,\...
- 15.6k
6
votes
0
answers
201
views
Smooth integers with lower bound on $\omega(n)$
Define $(b,c)$-smooth integers to be integers having all prime factors bigger than $c$ and smaller than $b$.
Probability a number is $(b,1)$-smooth is governed by the Dickman function while ...
- 1,826
6
votes
0
answers
211
views
some problems on sum of two squares
During my experiments with "Mathematica" I arrived to the following observations. My question is that are they interesting, known, solved or not. If they are known could you please give me a reference....
- 841
6
votes
0
answers
257
views
Convergence with the recurrence $T_{n+1}=T_n^2-T_n+\frac{n}{p_n}$
For each integer $n\geq 1$ I define the recurrence $$T_{n+1}=T_n^2-T_n+\frac{n}{p_n},$$
with $T_1=1$, where $p_k$ denotes the $k$-th prime.
So multiplying by $(-1)^n$ and telescoping gives that for ...
6
votes
0
answers
139
views
Upper bound for number of primes close to the next prime
Let $p_n$ denote the $n$th prime number and let $g_n := p_{n+1} - p_n$ be the $n$th prime gap. I'm looking for a good upper bound for the quantity
$$G(x, y) :=\#\{p_n \leq x : g_n \leq y\} ,$$
holding ...
- 113
6
votes
0
answers
206
views
Primes $p\in(n,2n)$ with $(\frac{-n}p)=-1$
Bertrand's postulate proved by Chebyshev states that for any $x>1$ there is a prime $p$ in the interval $(x,2x)$. In 2012 I considered some refinements of this by imposing additional requirement ...
- 15.6k
6
votes
0
answers
195
views
A recursion with a number-theoretic function
For a positive integer $Q$, let
$$ s(Q) := Q\,\sum_{p\mid Q} \frac1p, $$
where the sum extends over all prime divisors of $Q$; also, let $s(0)=0$. Thus, we have, for instance,
$s(1)=0$, while $s(p^\...
- 23k
6
votes
0
answers
266
views
On the determinant $\det[\sec2\pi\frac{jk}p]_{0\le j,k\le(p-1)/2}$
On the basis of my computation, I have the following conjecture involving the secant function.
Conjecture. Let $p$ be an odd prime and define
$$S_p:=\det\left[\sec2\pi\frac{jk}p\right]_{0\le j,k\le (...
- 15.6k
6
votes
0
answers
255
views
Gaussian square-free moat
Is there a sequence $\{z_n\}_{n=1}^\infty$ of distinct square-free
Gaussian integers with $$\sup_{n \geq 1} |z_{n+1} - z_n| < \infty ?$$
For the analogous problem with Gaussian primes instead, ...
- 11.3k
6
votes
0
answers
139
views
$p >2$ is a prime, any facts about congruence relation between the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$?
Let $p$ be an odd prime. This is a question about the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$,which we denote by $h(p)$ and $h(-p)$ respectively. While doing my research on number theory I came ...
- 363
6
votes
0
answers
664
views
Show this number always is composite number
Conjecture:
Let $m$ be a positive integer. Then
$$f(m)=(2m)^{2m+1}+m^{2m+1}\cdot (2m+1)^m+(2m+1)^{2m}$$
is not a prime number.
One can prove it when $m$ is odd number, it is clear that $f(m)$ ...
- 4,280
6
votes
0
answers
233
views
admissible tuples vs. prime tuples
Let $\rho^\ast(x)$ denote the maximal length of an admissible sequence in $[1,x]$, i.e. of a sequence which does not cover all the residue classes modulo any $n\geq 2$. Hensley and Richards (1974) ...
- 105k
6
votes
0
answers
435
views
Any ways to Simplify Daboussi's Argument for Prime Number Theorem?
One strategy to prove the Prime number theorem involves removing some factors:
$$ \limsup_{x \to \infty} \underbrace{\frac{1}{x}\sum_{n \leq x} \mu(x)}_{\color{red}{A}}\leq
\prod_{p \leq y} \left( 1 ...
- 22.8k
6
votes
0
answers
506
views
Prime gap counts in short intervals
Since it is conjectured that the twin prime count at $n\sim2 C_2\ \frac{n}{\log^2n},$ where $C_2 = \prod_{p\ge 3} \frac{p(p-2)}{(p-1)^2} = 0.66016 18158 \dots,$ it follows that the twin prime count ...
- 1,903
6
votes
0
answers
98
views
Prime divisors of the norm of the first coefficient of an elliptic newform at width-1 cusps.
Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ and let $f$ be its newform.
Suppose $p \geq 5$ is a prime such that $p^2 \mid N$. We assume $f$ is $p$-minimal, which is equivalent to that the ...
- 221
6
votes
0
answers
448
views
Are there always at least *five* divisions?
@JosephO'Rourke asked a question about a Collatz like function related to primes:
$f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is prime} \\
\lfloor n/2 \rfloor & \text{if} \;n \;\text{...
- 1,375
6
votes
0
answers
298
views
Can integers be distorted to make primes more regular?
Given a set $P$ of real numbers $\ge 1$, define the gap among different products in $P$ as
$$g(P) = \inf \big\{\prod_{i=1}^n p_i^{a_i} - \prod_{i=1}^n p_i^{b_i} \mid p_i\in P;\,\, p_i\ne p_j \,\text{ ...
- 5,103
6
votes
0
answers
260
views
A conjecture of Erdos on consecutive differences of primes
Let $d_k = p_{k + 1} - p_k$ be the difference between consecutive primes and define
\begin{equation}
e_k = \left\{\begin{array}{c l} 1 &, d_{k + 1} > d_k \\ 0 &, \text{otherwise} \end{array}...
- 338
6
votes
0
answers
505
views
$x^2+1$ attaining almost prime values
Iwaniec, using the linear sieve, proved that $n^2+1$ can be a product of at most two primes infinitely often and furthermore a lower bound of the correct order of magnitude for the number of such ...
- 3,062
6
votes
0
answers
322
views
Should I expect to see numbers this smooth?
I have a sequence $N_k$ of numbers whose growth I wish to determine, or at least
approximate nicely. When I look at the ratios of consecutive members,
I find some interesting simplifications ...
- 13k
6
votes
0
answers
726
views
Would the following conjectures imply Cramer's conjecture?
Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, (n-r,n+...
- 6,984
6
votes
0
answers
527
views
Any published references for this $O(n)$ time, $O(n^\epsilon)$ space identity for the count of primes?
I would like to know if this identity (or trivial equivalents) for $\pi(n)$, the count of primes, is currently published anywhere.
$D_{0,a}(n) = 1$
$D_{1,a}(n) = \lfloor n\rfloor-a-1 \ \ \ \ \ \ \ \ \...
- 627
6
votes
1
answer
1k
views
Arguments for the second Hardy–Littlewood conjecture being false?
Assume that $x,y > 2$, and that $x<y$. Then the second Hardy–Littlewood conjecture states that
$$\pi(x + y) - \pi(y) \leq \pi(x).$$
We can easily justify this heuristically, since
$$
\textrm{...
- 965
5
votes
0
answers
261
views
Primes generated by cyclotomic polynomials
Let $p$ be an odd prime, and let $f=\Phi_p$ be the $p$-th cyclotomic polynomial. Denote by $S_p$ the set of primes $q$ such that there exists a sequence of primes $p_1,\dots, p_g$ such that $p_1=f(1)=...
5
votes
0
answers
131
views
Taking integer values of a sequence of Beurling primes
Let $P=(p_j)_{j=1}^\infty$ be an increasing sequence of real numbers with $1<p_1$ and $\lim_{j\to\infty}p_j=\infty$. As mentioned in [1], Beurling proved that if the multiplicative group $N_P$ ...
- 51
5
votes
0
answers
83
views
Maximum size of difference sets with a bounded number of prime divisors
Call a subset $S\subset \mathbb{Z}$ $r$-smooth if the difference set $S-S$ contains numbers whose prime divisors lie in a set $P$ of distinct primes with $|P|=r$. Let $f(r)$ be the maximum size of any ...
- 4,862
5
votes
0
answers
541
views
Two conjectures for primes $p\equiv 1\pmod 8$
Motivated by my paper Quadratic residues and quartic residues modulo primes [Int. J. Number Theory 16 (2020), 1833-1858], here I pose two new conjectures for primes $p\equiv1\pmod8$ based on my ...
- 15.6k
5
votes
0
answers
251
views
Proofs in number theory that involve non-standard models of arithmetic
While reading an introductory text on model theory, I found it interesting that one can reformulate the famous conjectures about twin primes and Mersenne primes in terms of non-standard models of ...
- 273
5
votes
0
answers
160
views
Reducibility of $f(x)^{2^n}+1$ and $f(x)^{2^n}+g(x)^{2^n}$
Related to generalized Fermat numbers.
Let $f(x),g(x)$ be coprime polynomials with integer coefficients.
Assume that if $f(x)$ or $g(x)$ are of the form $h(x)^k$ then $k$ is power
of two.
Q1 Is it ...
- 25.4k
5
votes
0
answers
349
views
Smallest prime $p$ such that $2\mid\operatorname{ord}_p(q)$, the multiplicative order of $q$ modulo $p$
$\DeclareMathOperator\ord{ord}$Let $q$ be prime. I want to upper bound the smallest odd prime $p$ such that $2\mid\ord_p(q)$ (where $\ord_p(q)$ is the multiplicative order of $q$ modulo $p$).
Using ...
- 101
5
votes
0
answers
370
views
Large prime factors of n²+1
Iwaniec proved (and many people extended) that the number of $n \le x$ for which $n^2+1=P_2$ (product of at most two primes) is $\gg x/\log x$. I am wondering what is known/can be proved for the ...
- 51