Questions tagged [prime-numbers]
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
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Sums over primes in arithmetic progressions
Do we know anything about sums over primes in arithmetic progressions like the following:
$$\sum_{\substack{q \equiv a (\text{mod } l) \\ q \le x}} q^{\alpha}$$
where $q$ is a prime and $\alpha > ...
- 301
1
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0
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314
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From an inequality for the Euler's totient function to a combination of Firoozbakht's conjecture and Nicolas' criterion for the Riemann hypothesis
In this post we ask about the veracity of an inequality deduced from a combination of Firoozbakht's conjecture (see [1] or [2]) and Nicolas' criterion for the Riemann hypothesis (see for instance [3])....
3
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2
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216
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The graph and sign of $p_n-\operatorname{ali}(n)$, where $p_n$ is the $n$-th prime and $\operatorname{ali}(n)$ the inverse of the logarithmic integral
I'm inspired in [1] to ask the following question. My problem is that I have not an implementation of the inverse of the logarithmic integral $\operatorname{Li}(x)=\int_2^x\frac{dt}{\log t}$, that ...
4
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1
answer
282
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A similar lemma to a lemma due to Lagarias, for the partial sums of reciprocal of primes
I was inspired in Lemma 3.1 of [1] and in the Theorem 4.12 of [2] to ask about a similar statement that shows Lagarias in his paper as Lemma 3.1.
The Lemma from Lagarias's paper is that if $H(n)=\...
2
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1
answer
281
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Question on odd positive integers and Fermat factors
Let $n$ be an odd positive integer, Let $o=\operatorname{ord}_n 2$ be the order of 2 modulo $n$ and $m$ the period of $1/n, k$ is number of distinct odd residues contained in set $\{2^1,2^2,...,2^{n−1}...
- 359
4
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2
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438
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Sharp estimates for Meissel-Mertens constant
I wondered if it is possible to get a similar inequality like $(1.1)$ of Michael D. Hirschhorn, Approximating Euler's Constant, The Fibonacci Quarterly, Volume 49, Number 3 (August 2011) for the ...
0
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1
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218
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Is this theorem on the abundance of prime patterns/k-tuples known?
I am looking for references regarding the following statement.
For any two natural numbers x and y there must be a prime k-tuple (a, b, ...) corresponding to x consecutive primes (n+a, n+b, ...) for ...
3
votes
1
answer
413
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Primes from arithmetic and geometric progressions
The five primes, 131, 157, 211, 349, 739, are neither in arithmetic or geometric progression, but are instead the sum of the five corresponding terms of an arithmetic and geometric progression.
Are ...
- 3,677
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1
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338
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Is there a constant $\alpha$ such that: $P_{n+1} < P_n.\left(\frac{n+1}{n}\right)^\alpha$?
Is there a constant $\alpha$ such that:
$$P_{n+1} < P_n \cdot \left(\frac{n+1}{n}\right)^\alpha$$
Or
$$\lim_{n\to\infty}\frac{\ln\frac{P_{n+1}}{P_n}}{\ln\frac{n+1}{n}} < +\infty$$
Where $...
- 1,086
2
votes
2
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694
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Summation involving Euler's totient function
Does the following sum have a closed-form expression? I've tried an Inclusion-Exclusion interpretation, to no avail:
$f(n, p) = \sum_{i = 1} ^ n \lfloor \frac{n}{i} \rfloor \phi(p i) (-1) ^ i$
($p$ ...
2
votes
1
answer
228
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Equations involving arithmetic functions of primorials
Let $\sigma(n)=\sum_{1\leq d\mid n}d$ the sum of divisors, $\varphi(n)$ the Euler's totient function and we denote the primorial $\prod_{k=1}^n p_k$ as $N_n$, where $p_k$ denotes the $k$-th prime ...
7
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Does Morley's congruence characterize primes greater than $3$?
In 1895 Morley showed that $$\binom{p-1}{(p-1)/2}\equiv(-1)^{\frac{p-1}2}4^{p-1}\pmod{p^3}$$
for any prime $p>3$.
In 2009, I formulated the following conjecture concerning the converse of Morley's ...
- 14.5k
3
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2
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333
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For any integer $n>1$, there always exists at least one prime number $p$ with $n < p< n+\left(\ln\Big(\frac{n}{\ln n}\Big)+1\right)^2$
Question: Is the conjecture as follows true or false?
For any integer $n>1$, there always exists at least one prime number $p$ with
$$n < p< n+\left(\ln\Big(\frac{n}{\ln n}\...
- 1,086
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votes
0
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263
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On the infinity of $\{p\in \mathbb {N}:\exists n\in\mathbb{N}~p| \left \lfloor{r^n}\right \rfloor\}$
I've already asked this same question on MSE here, but didn't get much help, so I will try on this site as well.
For which $r\in\mathbb{R}$ is the set $\mathscr{P}_r=\{p \in \mathbb{P}:\ (\exists n\...
- 377
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1
answer
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Are there infinite many two sided prime numbers?
A prime number $p=\overline{a_na_{n-1}\ldots a_1a_0}$ is called a two sided prime number if its reverse representation $q=\overline{a_0a_1\ldots a_{n-1}a_n}$ is a prime number too.
Are there ...
- 312
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0
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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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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
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Published articles in journals about the Firoozbakht's conjecture, whose main goal or focus is the study of this conjecture
I would like to know what articles are in the literature about the known as Firoozbakht's conjecture, see the Wikipedia Firoozbakht's conjecture.
Question. What articles have been published in ...
2
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0
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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,$$ where $\pi$ denotes the prime counting function ...
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0
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Are there infinitely many primes $p$, positive integers $ k, n $ such that $1 \le n < p$ and $p^k > n.rad(p^{k+1}−n)$?
Among $168$ prime numbers in range $1$ to $10^3$, there are $84$ prime numbers $n$ such that: $p^k> n.rad(p^{k+1}−n)$ where $1 \le n<p$ and $k=2,3,4$. There are also $84$ prime numbers $n$ such ...
- 1,086
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0
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Guessing of $n$th prime from "super- regularized" product of primes
( I've been thinking about asking this for a long time . Though this is not rigorous; It can be thought of as heuristic or extraction of information from different viewpoint.)
We know "super-...
- 690
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1
answer
497
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Generalization of Wilson's theorem for prime tuples
We know that Wilson's theorem states the following :
$x$ is a prime if $(\frac {\Gamma(x)+1}{x})$ is an integer .
We can extend this to Twin primes as :
$x$ and $x+2$ is prime if $(\frac {4(\Gamma(...
- 690
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votes
1
answer
197
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applications of finding least quadratic nonresidue mod $p$?
I saw some papers from famous mathematicians (assuming GRH or without it) which are devoted to finding bound for least quadratic nonresidues modulo prime number $p$.
My question is that why it is so ...
- 841
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0
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On characterizations for Mersenne primes involving the sum of divisor function
In this post we denote the sum of positive divisors function of an integer $n\geq 1$ as $$\sigma(n)=\sum_{1\leq d\mid n}d.$$
Then a prime of the form $2^p-1$ is called a Mersenne prime. These are ...
2
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0
answers
131
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How many divisors of $\phi(m)$ do not divide $m-1$?
Lehmer's totient problem asks if there exists a composite number $m$ such that $\phi(m)$ divides $m-1$. Lower bounds on $m$ has been established but we do not know if a solution exists. Clearly, if we ...
- 4,638
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1
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A conjecture on primitive tenth roots of unity
QUESTION. How to solve my following conjecture involving primitive tenth roots of unity?
Conjecture. Let $\zeta$ be any primitive tenth root of unity. Then
$$\prod_{k=1}^{(p-1)/2}(\zeta-e^{2\pi ik^2/...
- 14.5k
3
votes
1
answer
206
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Races that involve odd semiprimes: a first statement or conjecture
While I am studying the famous article [1], in English this is Andrew Granville and Greg Martin, Prime Number Races, The American Mathematical Monthly, vol. 113, (2006), I wondered what about a race ...
1
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0
answers
111
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Upper bound for $\alpha_{n}$ from Mertens' third theorem
This question is a follow-up to About Goldbach's conjecture.
I would like to know if an unconditional upper bound for $\alpha_{n}$, defined as $n(N_{2}(n)-\dfrac{nN_{1}(n)}{P(n)})$ (where $N_{2}(...
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0
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99
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A problem in modular roots
We have three mutually coprime integers $r,t,M$ where $M\asymp K^{\frac12-2\epsilon}$ and $r,t\asymp K^{\frac14+\epsilon}$ holds with some fixed $\epsilon>0$ and $K>0$ is a large parameter. ...
- 13.7k
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1
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262
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On a problem that equates $\frac{\text{prime}-1}{\operatorname{rad}(\text{prime}-1)}$ with the sequence of primorials
We denote for integers $m>1$ the product of the distinct prime numbers dividing $m$ as $$\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(...
4
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0
answers
232
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On the values of $\prod_{k=1}^{(p-1)/2}(e^{2\pi i/12}-e^{2\pi i k^2/p})$ for primes $p>3$
In a recent preprint, I investigated
$$S_p(x):=\prod_{k=1}^{(p-1)/2}(x-e^{2\pi ik^2/p}),$$
where $p$ is an odd prime and $x$ is a root of unity.
Motivated by Question 337879 and Question 338325, ...
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1
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302
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Number of prime factors and density
Let $\mathbb{N}$ denote the set of positive integers. For $n\in\mathbb{N}$ let $\mathbf{P}_n$ be the set of all positive integers $k$ such that there are at most $n$ different prime numbers that ...
- 45.5k
2
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0
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A conjecture on crossing numbers related to primes
For a permutation $\sigma\in S_n$, its crossing number $\text{cr}(\sigma)$ is the number of pairs $\{i,j\}$ with $i,j\in\{1,\ldots,n\}$ such that
$$i<j\le\sigma(i)<\sigma(j)\ \ \text{or}\ \ \...
- 14.5k
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0
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On the connection between $\pi(x)-Li(x)$ and $\theta(x)-x$
Let $\pi(x)$ be the number of primes $p$ not exceeding $x, \theta(x) = \sum_{p\leq x} \log p$ and $Li(x)$ be the logarithmic integral.
Is it true that
$$\pi(x)-Li(x) = \theta(x) - x + O(x^{1/2}\log^{...
- 824
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0
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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 ...
3
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0
answers
672
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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
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0
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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 ...
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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 ...
2
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2
answers
267
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On values of $n\geq 1$ satisfying that for all primorial $N_k$ less than $n$ the difference $n-N_k$ is a prime number
In this post I present a similar question that shows section A19 of [1], I was inspired in it to define my sequence and question. I am asking about it since I think that the problem that arises from ...
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0
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89
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Is the ratio of a number to the variance of its divisors injective?
The variance $v_n$ of a natural number $n$ is defined as the variance of its divisors. There are distinct integer whose variances are equal e,g. $v_{691} = v_{817}$. However I observed that for $n \le ...
- 4,638
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1
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99
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On the integral $I_s = \int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$-follow up question
This is a follow up on On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$
According to the answer that i got, $I_s$ is not known to converge for any real $s<1$. But suppose $I_s$ ...
user140392
2
votes
2
answers
428
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On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$
Define $\pi(x)$ to be the prime counting function and Li(x) the logarithmic integral. Let $I_s$ be defined as above.
Is $I_s$ known to be convergent for any real number $s<1$ ?
user140392
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0
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83
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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 ...
4
votes
2
answers
679
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On the product $\prod_{k=1}^{(p-1)/2}(x-e^{2\pi i k^2/p})$ with $x$ a root of unity
Let $p$ be an odd prime. Dirichlet's class number formula for quadratic fields essentially determines the value of the product $\prod_{k=1}^{(p-1)/2}(1-e^{2\pi ik^2/p})$. I think it is interesting to ...
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29
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2
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Is there a Kolmogorov complexity proof of the prime number theorem?
Lance Fortnow uses Kolmorogov complexity to prove an Almost Prime Number Theorem (https://lance.fortnow.com/papers/files/kaikoura.pdf, after theorem $2.1$): the $i$th prime is at most $i(\log i)^2$. ...
- 13.7k
2
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0
answers
165
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What about series involving strong primes?
I know about the importance in analytic number theory of the sutdy of series involving prime numbers or constellations of prime numbers, for example, if I am not wrong, major theorems are Mertens' ...
9
votes
1
answer
445
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A conjectural formula for the class number of the field $\mathbb Q(\sqrt{-p})$ with $p\equiv3\pmod8$
Question. Is my following conjecture new? How to prove it?
Conjecture. Let $p>3$ be a prime with $p\equiv3\pmod 8$, and let $h(-p)$ denote the class number of the imaginary quadratic field $\...
- 14.5k
3
votes
0
answers
284
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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 ...
0
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1
answer
448
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An upper bound for $\sqrt{p_{n+1}}$
Let $C$ be a positive constant. Is it true that for all sufficiently large integers $n$ the inequality $$\prod_{i=1}^n (1+\frac{1}{\sqrt{p_i}})>C\sqrt{p_{n+1}}$$ holds? (Here with $p_k$ is denoted ...
2
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1
answer
321
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Primes in shifted geometric sequence
Call a pair of integers $(a,b)$ trivial if it satisfies some simple divisibility condition, like for some prime $p$ we have $p$ divides both $a-1$ and $b+1$, or that $p$ divides both $a$ and $b$. This ...
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