All Questions
Tagged with nt.number-theory prime-numbers
518 questions with no upvoted or accepted answers
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Existence of minimal bases in additive combinatorics
Let $\mathbb{N}$ denote the set of natural numbers, including zero. A subset $X \subseteq N$ is a basis if $X + X = \mathbb{N}$. Clearly, if $X$ is a basis and $X \subseteq Y$, then $Y$ is also a ...
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122
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Convergence of a series related to counting distinct prime factors
I am here to ask whether the following series is convergent for all real $z$. I am also asking whether this is everywhere real analytic. I conjecture that it is convergent for all real input, or at ...
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355
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On a Duality between Riemann-weil explicit formula and Abel- Plana summation of trigonometric prime counting function:
Consider the analytic function $g(x)$
Now define
$f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$
Such that
$|f(x+it)|=o(e^{2πt})$
uniformly for every $x$...
- 790
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106
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Primes of the form power of 2 plus a prime
By Bertrand' postulate, there exists a prime between $2^n$ and $2^{n+1}$.
For every $n$, is there a prime $p < 2^n$ such that $2^n+p$ is a prime?
The smallest such primes are listed in OEIS A056206....
- 483
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151
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Mertens' Third Theorem for primes of the form $4n+1$
I am looking for upper and lower bounds for the following expression:
$$\prod_{\substack{p\le n \\ p \equiv 1\ mod\ 4}} \frac{p-1}{p}$$
Apart from the trivial one:
$$\prod_{\substack{p\le n \\ p \...
- 193
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101
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Sequence $a_1,a_2,\ldots$ with $a_j\in\lbrace 1,2,\ldots,j\rbrace$ such that almost all $\sum_{j=1}^na_j\cdot j!$ are prime-numbers
Are there prime-numbers having infinite left-expansions of non-zero coefficients in the factorial number system involving only prime numbers?
The question is really in the title : Is there an infinite
...
- 17.9k
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107
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Clumps of small multiples of large squares
Am I right to be surprised by this big clump of numbers divisible by large squares within a not-so-long interval? If so, should I be surprised because $(1)$ this rarely happens, or because $(2)$ it's ...
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146
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Remainder-balancedness of primes
Let $\mathbb{N}_+$ denote the set of positive integers. Consider the remainder function $\text{rem}:\mathbb{N}_+\times \mathbb{N}_+ \to \mathbb{N}\cup\{0\}$ defined by $$(n,d) \mapsto n - \Big(\Big\...
- 48.9k
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110
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What will be the set of non-Wieferich numbers if the set of non-Wieferich primes is finite?
Integer $n$ is Wieferich number if $2^{\phi(n)}-1 \equiv 0 \pmod {n^2}$.
Wieferich prime is Wieferich number with $n$ prime.
It is an open problem if there are infinitely many Wieferich primes
and ...
- 25.4k
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68
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Around similar inequalities than an inequality due to Nicolas, that involve products of consecutive Ramanujan primes
This is cross-posted (and this post is a version to ask just around the veracity of Conjecture 1) as the post with identifier 3594907 and same title), that I've edited on Mathematics Stack Exchange ...
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462
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Relation between sieve wheel and Sundaram sieve
I made this sieve for prime numbers, which I briefly describe:
We consider $\quad p=r+modulus \cdot k \quad$ with $\quad modulus=p_1*p_2* \cdots *p_m$
and then we choose an appropriate reduced ...
- 179
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136
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Bounded sums involving primes
I'm trying to generalize the Theorem 2.7.1 in [1] where they prove:
$$\sum_{p \leq x} f(p) = \int_{2}^{x} \frac{f(t)}{\log{t}} dt + \epsilon(x)f(x) - \int_{2}^{x} \epsilon(t) f^{'}(t) dt $$
where $\...
- 109
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113
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If we weaken Polignac's conjecture to an existential claim, can it be proved?
Polignac's conjecture (unproved) states that, for any integer $k \geq 1$, there exist infinitely many $p$ such that $p$ and $p+2k$ are both prime. Suppose that we weaken the consequent to require only ...
- 459
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On $\sum_{\rho\in D} \text{dist}(\rho)=\frac{1}{2\pi i} \int_{\partial{D}}\log \zeta(s)\ ds$
Let $D$ denote a closed two dimensional figure as: $D=2+iT\to 2\to 2-\delta\to 2-\delta+i(T-\delta)\to \frac{1}{2}+\epsilon+i(T-\delta)\to\frac{1}{2}+\epsilon\to\frac{1}{2}-\epsilon\to \frac{1}{2}-\...
- 11
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91
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Reducing the number of terms in Waring-Goldbach problem by allowing exponents to vary
Assuming the Waring-Goldbach problem (see https://en.m.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem) has a positive solution, can we reduce the number of terms $t$ to some value $t'$ by allowing ...
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Relevance of the deduction of similar theorems than Maier's theorem for other prime constellations
A year ago I asked this question on Mathematics Stack Exchange with identifier 4245823 and same title Relevance of the deduction of similar theorems than Maier's theorem for other constellations of ...
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A similar inequality for the Dedekind psi function, than an inequality stated by Schinzel
I would like to ask about the next question that seems to me interesting. I know an article that was written by Andrzej Schinzel in which he stated Lemma 2. In this post we denote the Dedekind psi ...
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144
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better estimates than the prime number Theorem in Euclidean domains
For a unique factorization domain we know that we have some the analogues of fundamental theorem of arithmetic, and can build elements by using 'building blocks'. For me the easiest examples are ...
- 1,561
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197
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Closed form for sum of modulo remainders
Given a set of natural numbers {$N_1, ...N_n$} and another natural number $N$, find a closed form solution for the below sum
$M = \sum_{i=1}^n N_i \% N$
where $N_i, N, n \in \mathbb N$
and $...
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150
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Primality of $\ \frac{b^p-a^p}{b-a}$
Can you prove the following:
Conjecture: Let $\ p\in\mathbb P\ $ be an arbitrary prime. Then there exist two relatively prime integers $\ a\ $ and $\ b\ $ such that $\ a>0\ $ and $\ b>1\ $ and
$...
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177
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Is there a link between Elliott-Halberstam and weak Hardy-Littlewood-Goldbach conjectures?
Let $\theta$ be such that $EH(\theta)$ holds, where $EH$ stands for Elliott-Halberstam. Can one get an explicit lower bound $\delta_{\theta}$ for the quantity $\delta$ appearing in the weak Hardy-...
- 6,984
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257
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Unexpected autocorrelations in sequence of primes modulo 4
It is well known that there is a little bias in the distribution of prime residues modulo 4. But the bias eventually vanishes. I looked at the first million primes, and the counts are as follows:
...
- 3,098
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102
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On simple examples of unimodularity
$w=z=x+ 1 =y−1$ provides $wz−xy=w^2−(w−1)(w+ 1) = 1$. Hence if $x,y$ are odd then $w,z$ are even and all four integers are close.
Is there elementary example where only $w$ is even and all four ...
- 13.9k
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135
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On a deterministic primes search problem
I feel the following problem might be resolved already. But I could not find any related answers.
If $p_1,p_2,\dots,p_t$ are primes where $2\leq t=o(\log n)$ is there a prime within $$\prod_{i=1}^...
- 13.9k
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118
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what are all possible pairs (k,m) such that n=2k^2+ m^2
I am working on a problem in number theory and would like to count all possible ways to partition an integer $n\geq 1$ into pairs $(k,m)$ of positive integers such that $n=2k^2+m^2$ and $n=4k^2+m^2$. ...
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112
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The number of solutions of $2^xpx+k=y^2$
Let's consider the family of diophantine equations
$$2^xpx+k=y^2$$
being $p\gt2$ a prime and $k$ a positive integer.
An example is given by the equation
$$2^x\cdot3x+97=y^2$$
that presents, at least, ...
- 1,008
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62
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On complexity of a particular prime problem
Is the following problem in $PH$ and is it complete for any class?
Problem: Is the $i$th bit of the $m$th prime $1$?
It appears to require a counting quantifier which has to demonstrate witness is the ...
- 13.9k
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118
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Primes in many variables polynomials form
As in this question >> Primes of the form $x^2 + y^2 + 1$ There are $\asymp \dfrac{x}{\log^{3/2} x}$ primes of the form $a^2+b^2+1$ up to $x$. I read a book stated that there exist a polynomial $...
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On a generalised result of Mertens
Let $\varphi$ be the Euler totient function, $N_k$ denote the product of the first $k$ primes and define $$f(k, r) = \frac{(N_k)^r}{\varphi((N_k)^r) \log \log ((N_k)^r)}.$$
where $r \in \mathbb{N}$. ...
- 1,019
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123
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Testing the primality of Mersenne and Fermat numbers using third order recurrence relation
Can you prove or disprove the claims given below?
Inspired by generalization of Lucas-Lehmer test I have formulated the following claims:
Claim 1 Let $M_p=2^p-1$ where $p$ is an odd prime number , ...
- 2,661
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100
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Is there always a prime between $\beta z^\alpha$ and $\beta (z+1)^\alpha$ for $z$ large enough, and a fixed $\alpha < 2$?
I am interested if possible in $\beta = \frac{2}{3}, \alpha=\frac{3}{2}$ and $z$ is a positive integer or real number. My choice here is related to some progress I make in additive combinatorics (see ...
- 3,098
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133
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A quadratic trinomial that generates only prime numbers of the form $4m+1$
It is known that Euler's polynomials $\,n^2+n+p\,$ ($p\,$ prime) represent a prime for $\,n=0,\,...,\,p-2\,$ if and only if the field $\,Q (\sqrt{1-4p})\,$ has class number $\,h=1$.
The best ...
- 1,008
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143
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Given $\,m=\prod_k {p_k}^{\alpha_k}\,$ and the function $\,g(m)=\sum_k \alpha_k(p_k-1)^2$, find all solutions of the equation $\,g(2n)=n$
Let's consider the unique decomposition of a natural number $\,m\,$ into its prime factors:
$$\prod_k {p_k}^{\alpha_k}$$
Then, let's define the following arithmetic function (completely additive) $\,g:...
- 1,008
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Can power of a prime number be approximated by product of powers of two adjacent prime numbers?
With prime numbers $a < b < c$ and no primes exist in ranges $(a, b)$ and $(b, c)$, is it possible that there exists positive integers $x$, $y$, $z$ such that $|a^x c^z-b^y|=2$?
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Inequalities $\pi(x^a+y^b)^\alpha\leq \pi(x^c)^\beta+\pi(y^d)^\gamma$ involving the prime-counting function, where the constants are very close to $1$
Let $\pi(x)$ be the prime-counting function, I'm curious about if a suitable variant of the second Hardy–Littlewood conjecture (this corresponding Wikipedia)
$$\pi(x^a+y^b)^\alpha\leq \pi(x^c)^\beta+\...
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133
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Is this weaker form of Chowla's conjecture potentially interesting?
I began to read a post on Terry Tao's blog dealing with Chowla's conjecture, and doing so I came to think about a weaker form thereof.
So for a given positive integer $m$, let $S_{m}(x)$ denote the ...
- 6,984
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118
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At least a prime k-tuple
Let $k \in\mathbb{N}, k \geq 2$, and $\mathbb{P}$ represent the set of prime numbers.
Consider the k-tuple $(0,h_1,h_2,\cdots,h_{k-1})$ with $0 < h_1 < \cdots < h_{k-1}$.
The well known k-...
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Is it possible to get a conjecture similar to Mandl's conjecture for a different arithmetic function of number theory, mainly related to primes?
I'm curious to know if are in the literature analogous conjectures to the conjecture due to Mandl, I ask about these analogous conjectures for different sequences playing an important role in number ...
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128
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Number of primes skipped by binomial coefficients?
Take $$B(l,n)=\binom{n+l}{n}$$ and $\mathcal P(t)=\{p\mbox{ prime}:p|t\}$.
What is the cardinality of $\mathcal P(B(l,n))$?
What is minimum cardinality of $L\subseteq\{1,\dots,n\}$ such that $$\...
- 13.9k
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434
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Matrix sieve algorithm
I proposed "matrix sieve" algorithm for finding primes as two pairs of 2-dimensional arrays:
positive integers which do not appear in these
arrays are indexes $k$ of primes in the sequences $S1(k)=...
- 11
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65
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Is the fundamental partition associated to $n$ the partition of $r_{0}(n)$ in $k_{0}(n)$ parts that maximizes entropy?
As usual, under Goldbach's conjecture, let's define for a large enough composite integer $n$ the quantities $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$ and $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-...
- 6,984
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759
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On sets of coprime integers in intervals
Briefly,
Question: Is it "good enough" to use least prime factor in choosing a maximal set of coprime integers in an interval?
The post title comes from a 1993 paper of Erdos and Sarkozy. They ...
- 13k
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116
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Reference request for bounds of $n$-th composite
Motivation
I will briefly elaborate here my motivations for asking the question. If you are not interested in it then please go to the questions.
Recently during trying to understand and prove the ...
user57432
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115
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What is known about the lower bound for the integers $n$ for which $n$ minus the first $k$ odd primes are $k$ composite numbers?
Question edited in view of the comments below
By Yamada's paper we can conclude that if $n>e^{e^{36}}$ be an even number then it can always be written as the sum of a prime and a semi-prime.
My ...
user57432
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161
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Counting divisors of primes plus or minus 2
I was naively exploring the (usually) composites $\pm 2$ from a
prime $p$, wondering if there might be some asymmetry,
and made this histogram of the difference $\Delta$ in
the number of divisors of $...
- 151k
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142
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Mobius function on values of an irreducible quadratic polynomial
Are there infinitely many integers $n$ for which $n^2 + 1$ is square-free, and has an even number of (necessarily distinct) prime factors ?
- 11.3k
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149
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On primitive roots of the form $5^k+10^m$ with $k$ and $m$ nonnegative integers
Let $p$ be any prime. It is well known that the set
$$G_p=\{0<g<p:\ g\ \text{is a primitive root modulo}\ p\}$$
has cardinality $\varphi(p-1)$, where $\varphi$ is Euler's totient function. It is ...
- 15.6k
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143
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A question about the Heilbronn-Rohrbach Inequality
Let $\Phi(x,p)$ = the number of integers $i$ where $0 < i \le x$ and $\gcd(x,p)=1$.
Let $p\#$ be the primorial for $p$.
Using the inclusion-exclusion principle:
$$\Phi(x,p\#) = \sum_{d|p\#}\mu(d)...
- 1,049
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230
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Factorial : Gamma :: Primorial :?
Is there a unique function with the following properties:
f is meromorphic on the complex plane;
f is log-convex for n ≥ 1
$f(n) = n\#$ for n prime and ≥ 2, where # is the primorial function, and $f(...
- 309
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Bound for $|p_n - \operatorname{li}^{-1}(n)|$
It is well-known that $|\pi(x)-\operatorname{li}(x)| \leq \epsilon(x)$, where $\pi(x) = \sum \limits_{p \leq x} 1$ is the prime counting function, where $\operatorname{li}(x) = \int \limits_{2}^{x}\...
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