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.
1,944
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Are the binary digits of the sequence of the prime numbers correlated?
Let $p_n\geq 3$ be the $n$th prime number with the binary expansion $p_n = \sum_{k=0}^{\infty} b_{nk}2^k$ ($b_{nk}\in\{0,1\}$). Let's write $q_{nk} = 1-2b_{nk}$.
Question: Is it true that for $k,l\...
- 1,755
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107
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Counting prime factors of polynomial functions
Let $\Omega(n)$ denote the number of prime factors (counted with multiplicity) of a non-zero integer $n$. For $f \in \mathbb Z[X]$ non-zero, let $$m(f) = \liminf_{n \to \infty} \Omega(f(n))$$
(1) Is $...
- 11.7k
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128
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On distribution of prime pairs coming from certain polynomials
Consider the polynomials
$$g(x)=(2x)^4+((2x)^2+1)^2$$
$$h(x)=(2x)^4+((2x)^2-1)^2.$$
If $k$ odd integers $x_1,\dots,x_k$ are uniformly randomly chosen in $(t,2t)$ and the polynomials are evaluated at ...
- 13.4k
4
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279
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How to explain this number-theoretic seeming “almost coincidence”?
For natural numbers $n\geq2$, let $d(n)$ be the number of divisors of $n$, and let
\begin{equation}
g(n)=n\sum_i r_i(p_i-1)
\end{equation}
where $n=\prod_i p_i^{r_i}$ is the factorisation of $n$ as a ...
- 141
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0
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76
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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....
- 435
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2
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135
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Calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$ [closed]
How to calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$, where $n$ and $m$ are positive numbers.
We guess that: the great common factor is $1$.
- 557
3
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1
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194
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Bins-and-primes (prime divisors of $\prod(a_i-a_j)$, II)
This is a refined version of a question I have recently posted.
For a prime $p$, let $\varphi_p\colon\mathbb Z\to\mathbb Z/p\mathbb Z$ denote the canonical homomorphism from the integers onto the ...
- 22.2k
2
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1
answer
142
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Prime divisors of $\prod(a_i-a_j)$
For a prime $p$, let $\varphi_p\colon\mathbb Z\to\mathbb Z/p\mathbb Z$ denote the canonical homomorphism from the integers onto the group of order $p$.
Given an integer $n\ge 3$, what is the smallest ...
- 22.2k
1
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41
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Convergence of Farey series integral of a "density" function as the order tends to infinity
Let $F_n$ denote the $n$-th Farey sequence, and let $q$ be a rational number such that $0 \leq q \leq 1$. I am studying the convergence of a specific integral related to Farey series, defined as ...
- 367
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85
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Conjectured probable primality test for generalized repunits numbers $\frac{b^p-1}{b-1}$
Let $R_p = \frac{b^p-1}{b-1}$ when $b$ is a number $\ge 3$ and not a perfect power (ex : $3^2$, $6^2$, $5^3$ ...) and $p$ a prime number $\ge 5$.
Let the sequence $S_i = 2 \cdot (S_{i-1}+2)^b-2$ with $...
- 1
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134
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Asymptotics for $\mathrm{Zi}(x)$ and comparison to $\mathrm{Li}(x)$
Consider a function that attempts to count primes
$$\mathrm{Zi}(x)=\frac{1}{e}\sum_{k=1}^\infty\frac{(\ln x)^k}{kk!\phi(k)}$$
where
$$ \phi(k)= \sum_{n=1}^\infty e^{-n^k} $$
Based on some preliminary ...
2
votes
1
answer
119
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Estimating the minimum number of distinct least prime factors found in range of $c$ consecutive integers
When I look at the count of distinct least prime factors for a range of consecutive integers, I am seeing the same minimum number appear again and again. I am wondering if this number represents the ...
- 1,019
1
vote
1
answer
226
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Upper bound of number of prime factors
If I'm given a prime number $p$: is there an upper bound to the number of prime factors of $p−1$? Alternatively, is there a way to calculate the number of prime factors of $p−1$ without actually ...
- 13
5
votes
2
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274
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A prime divisor $p$ of Fermat number $F_n$ is a Wieferich prime if and only if $p^2$ divides $F_n$ [closed]
Let $F_n=2^{{2^n}}+1$, $n\geq 1$ ( Fermat numbers) and $p>2$ a prime number sucht that $p|F_n$
I want to show if true that :
$p$ is Wieferich prime number $\Longleftrightarrow $ $p^2|F_n$
the ...
- 1,503
83
votes
2
answers
5k
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A little number theoretic game
I came up with this little two player game:
The players take turns naming a positive integer. When one player says the number n, the other player can only reply in two different ways: They can either ...
- 761
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votes
1
answer
292
views
What heuristic arguments support Montgomery's conjecture for primes in short intervals?
I have a question regarding a conjecture due to H. L. Montgomery on the number of primes in short intervals. The conjecture apparently arises from probabilistic reasoning upon assuming the Riemann ...
- 405
11
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2
answers
1k
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Mertens-like theorem
Mertens' first theorem states that
$$
\sum_{p \leq n} \frac{\log p}{p} = \log n + O(1).
$$
I read in this paper that the following variant is "classical":
$$
\sum_{p \leq n} \frac{\log p}{p -...
6
votes
0
answers
157
views
$p^2+a^2$ can be a squarefree number with all prime divisors less than $p$?
Let $p$ be a prime $\ge 31$.
Is there an integer $a < p$ such that $p^2 + a^2$ is a squarefree and all of its prime divisors are less than $p$?
For example, for $p=31$, $31^2+5^2 = 986 = 2 \times ...
- 435
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127
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Mertens' Third Theorem for primes of the form $4n+1$
I am looking for upper and lower bounds for the following expression:
$$\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 \...
- 163
5
votes
1
answer
299
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About an asymptotic behavior in number theory
Where can I read about the asymptotic behavior (with $N$ tending to infinity) of the sum of the fractional parts obtained from dividing $N$ by all prime numbers up to $N$ divided by the number of ...
3
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0
answers
145
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What is the density of numbers which have at least two divisors whose sum is a perfect square?
Note: This question was posted in MSE about two years ago but it not receive an answer. Hence posting in MO.
A positive integer is said to have square-sum divisors if it has at least two divisors ...
- 4,416
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0
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99
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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
...
- 16k
1
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0
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137
views
Can every average of twin prime pairs be expressed by the sum of two smaller averages
Let's call $A$ the group of Averages of twin prime pairs (the composite between two twin primes as in OEIS A014574).
I noticed that for small numbers in $A$ they can be expressed as the sum of two ...
2
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0
answers
190
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Conjecture about primes and Fibonacci numbers
I posted this conjecture on math.stackexchange, but I received no answer proving or disproving it: if $ m > 4 $ is a positive integer not divisible by $ 2 $ or $ 3 $, it's ever possible to find a ...
- 131
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603
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Mysterious sum equal to $\frac{7(p^2-1)}{24}$ where $p \equiv 1 \pmod{4}$
Consider a prime number $p \equiv 1 \pmod{4}$ and $n_p$ denotes the remainder of $n$ upon division by $p$. Let $A_p=\{ a \in [[0,p]] \mid {(a+1)^2}_p<{a^2}_p\}$.
I Conjecture
$$\sum_{n \in A_p } n=\...
- 1,503
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0
answers
226
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A set of sequences and their relations among each other
Define an order $o_p(e)$ as follows: for each even $e$, get a unique sum $p+q$ of primes $p$, $q$ (if there is one). Choose $p$ such that $p$ is the least prime in the list of primes that have been ...
2
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0
answers
253
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Euler's totient function and primes
I'm looking for a proof of this conjecture: https://math.stackexchange.com/questions/4478597/eulers-totient-function-and-primes
$\phi$ denotes the Euler's totient function, $a$ denotes a natural ...
- 29
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1
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334
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Recent works on the Hardy-Littlewood conjecture on primes represented by quadratic polynomials
I have been working on my master's thesis which is about the equivalence of the Hardy-Littlewood conjecture on primes represented by quadratic polynomials and the Lang-Trotter conjecture for CM ...
- 239
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0
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74
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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\}...
- 28.7k
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1
answer
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Primality of a number of more than 50k digits
With modern tecnology is it possible to prove the primality of a number of more than 50k digits?
Obviously not a prime for which specific methods for testing primality are known like Mersenne primes.
- 147
6
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1
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454
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Some conjectures about prime gaps
I checked some relations between primes, here $1<n<10^5$ and $p_n$ is the $n$th prime.
$a) p_n^{1/3} - p_{n-1}^{1/3}<1/2$
$b) p_n^{1/n} - p_{n-1}^{1/n}<1/n $
$c) (\log p_n)^{1/2} - (\...
- 549
1
vote
1
answer
249
views
Is $p_1p_2\ldots p_n +1$ a prime number for infinitely many $n\in \mathbb{N}$? [duplicate]
Let $p_1,p_2,\ldots,p_n,\ldots,$ be the sequence of prime numbers. Are there infinitely many $n\in \mathbb{N}$ such that the natural number $p_1p_2\ldots p_n +1$ is a prime number?
- 139
25
votes
0
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489
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Elliptic analogue of primes of the form $x^2 + 1$
I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
- 12.8k
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What are examples of problems we know how to solve for primes (or prime powers), but not for composites?
I am interested in seeing examples of research problems which fall into one of the two following categories:
A problem which is solved in the case of primes (or prime powers), but which remains open ...
2
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0
answers
171
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Is the Goldbach conjecture easier if we allow 1 as a prime?
I hope this is the right site for the question.
Is the Goldbach conjecture easier if we allow 1 as a prime? (12=1+11 would be allowed as Goldbach sum for 12)
IOW: if we can prove Goldbach for the case ...
- 129
1
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0
answers
176
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Getting rid of complex zeros of function with zeros the primes?
From our Note: simple real function with zeros greater than one the primes
simple real function with zeros greater than one the primes:
$j_1(x)=(\sin(\pi x))^2+(\sin(\pi \frac{\Gamma(x)+1}{x}))^2$.
...
- 23.9k
1
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0
answers
58
views
Set from a diophantine equation with similar statistics to primes
While doing some computational calculations with some diophantine equations, I came across with some sequences from solutions of quartic and quintic equations with slowly decreasing frequency, similar ...
1
vote
1
answer
140
views
Lucas–Lehmer test and triangle of coefficients of Chebyshev's
In the Lucas–Lehmer test with $ \quad p \quad $ an odd prime.
we know that $ \quad S_0=4 \quad $ and $ \quad S_i=S_{i-1}^2-2 \quad $ for $\quad i>0 \quad$
$M_p=2^p-1 \quad$ is prime if $ \quad S_{p-...
- 179
3
votes
1
answer
367
views
Curious infinite product, convergence, connection to prime numbers
I have been playing with the following function:
$$
f(x)=\frac{\pi x (1-x^2)}{\sin\pi x}\prod_{k=2}^\infty \frac{\sin(\pi x/k)}{\pi x/k}
$$
It is hard to get correct numerical values. I'll start with ...
- 2,986
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votes
0
answers
77
views
Clumps of small multiples of large squares
Am I right to be surprised by this big clump of numbers divisible by large squares within a not-so-long interval? If so, should I be surprised because $(1)$ this rarely happens, or because $(2)$ it's ...
- 10.3k
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votes
0
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94
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Prime races in two competing arithmetic progressions - error bound
I read an article by Andrew Granville on the subject, there's actually quite a bit of recent literature on the topic. My problem is as follows. I have two sequences of primes: $(p_{1,n})$ and $(p_{3,n}...
- 2,986
4
votes
1
answer
358
views
Primes of the form $d^2+d+1$
Is $d^2+d+1$ prime for infinitely many $d\in \mathbb{Z}_{>0}$?
This is expected by the Bunyakovsky conjecture which says that, under some conditions, given a polynomial $p(x) \in \mathbb{Z}[x]$ we ...
- 348
3
votes
1
answer
312
views
Generating prime $\ p_{n+1}\ $ (the complete version)
Let $\ p_n\ $ be the consecutive primes starting with
$\ p_0:=2.\ $ Let $\ M(n)\ $ be the multiplicative monomial
generated by $\ \{p_k:\ k=0\ldots n\}\ $ (of course $\ 1\in M(n)$).
Could you prove or ...
- 4,325
0
votes
0
answers
143
views
Remainder-balancedness of primes
Let $\mathbb{N}_+$ denote the set of positive integers. Consider the remainder function $\text{rem}:\mathbb{N}_+\times \mathbb{N}_+ \to \mathbb{N}\cup\{0\}$ defined by $$(n,d) \mapsto n - \Big(\Big\...
- 42.6k
-5
votes
1
answer
153
views
It is known if $n$ is prime for all $n\leq N$ [closed]
This is more of a curiosity than a research question, but I could not find it answered anywhere. What is the largest $N$ for which the statement in the title is true? I have recently read that the ...
- 305
2
votes
2
answers
232
views
Inequalities for two functions related to the primorial function
Added: As remarked in the answers below, my question has a negative (and well-known) answer.
We denote by $\mathcal P=\lbrace 2,3,5,7,\ldots\rbrace$ the set of prime-numbers and by
$\mathcal P^*=\...
- 16k
2
votes
1
answer
114
views
Weak Siegel–Walfisz property
Let $f:\mathbb N \to \mathbb C$ be an arithmetic function. There are various ways to define what the Siegel–Walfisz (S–W) property is for $f(n)$. One simple way is that
there exists some function $g(...
- 2,862
0
votes
0
answers
29
views
'$r$-prime factor number' distribution centering an '$l$-prime factor number'
Let '$k$-prime factor integer' $n$ be an integer of form
$$2^t\prod_{i=1}^{k'} p_i^{e_i}$$
where $1\leq k'\leq k$ and at every $i\in\{1,\dots,k\}$ $p_i$ is a distinct odd prime and $e_i\in\mathbb Z_{&...
- 13.4k
14
votes
0
answers
284
views
An 'onion-structure' for roots of a series associated to prime numbers?
The series $$\sum_{n=1}^\infty\frac{z^{p_n-n}}{n!}$$ associated to the
sequence $p_1=2,p_2=3,p_3=5,p_4=7,p_5=11,\ldots$ of prime numbers
defines a holomorphic function in the open disc of radius $e$.
...
- 16k
2
votes
0
answers
58
views
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?
- 13.4k