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.
2,022
questions
1
vote
0
answers
52
views
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 ...
- 363
2
votes
1
answer
136
views
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 ...
- 1,039
2
votes
1
answer
496
views
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 ...
- 23
5
votes
2
answers
343
views
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
84
votes
2
answers
6k
views
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 ...
- 771
3
votes
1
answer
366
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
votes
2
answers
1k
views
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":
$$
\sum_{p \leq n} \frac{\log p}{p -...
6
votes
0
answers
166
views
$p^2+a^2$ can be a squarefree number with all prime divisors less than $p$?
Let $p$ be a prime $\ge 31$.
Is there an integer $a < p$ such that $p^2 + a^2$ is a squarefree and all of its prime divisors are less than $p$?
For example, for $p=31$, $31^2+5^2 = 986 = 2 \times ...
- 483
0
votes
0
answers
143
views
Mertens' Third Theorem for primes of the form $4n+1$
I am looking for upper and lower bounds for the following expression:
$$\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 \...
- 183
5
votes
1
answer
325
views
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
votes
0
answers
158
views
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 ...
- 4,638
0
votes
0
answers
100
views
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
...
- 17.4k
1
vote
0
answers
145
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
votes
0
answers
262
views
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 ...
- 219
18
votes
0
answers
663
views
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
1
vote
0
answers
229
views
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 ...
1
vote
0
answers
281
views
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 ...
- 169
4
votes
1
answer
430
views
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 ...
- 311
6
votes
0
answers
109
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\}...
- 30.6k
12
votes
1
answer
2k
views
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.
7
votes
1
answer
474
views
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} - (\...
- 529
1
vote
1
answer
256
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?
- 312
26
votes
0
answers
526
views
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 ...
- 13.1k
11
votes
9
answers
1k
views
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
votes
0
answers
199
views
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
vote
0
answers
179
views
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$.
...
- 24.2k
1
vote
0
answers
61
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
145
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
418
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}
$$
It is hard to get correct numerical values. I'll start with ...
- 3,088
0
votes
0
answers
91
views
Clumps of small multiples of large squares
Am I right to be surprised by this big clump of numbers divisible by large squares within a not-so-long interval? If so, should I be surprised because $(1)$ this rarely happens, or because $(2)$ it's ...
- 11.9k
0
votes
0
answers
100
views
Prime races in two competing arithmetic progressions - error bound
I read an article by Andrew Granville on the subject, there's actually quite a bit of recent literature on the topic. My problem is as follows. I have two sequences of primes: $(p_{1,n})$ and $(p_{3,n}...
- 3,088
4
votes
1
answer
375
views
Primes of the form $d^2+d+1$
Is $d^2+d+1$ prime for infinitely many $d\in \mathbb{Z}_{>0}$?
This is expected by the Bunyakovsky conjecture which says that, under some conditions, given a polynomial $p(x) \in \mathbb{Z}[x]$ we ...
- 348
3
votes
1
answer
319
views
Generating prime $\ p_{n+1}\ $ (the complete version)
Let $\ p_n\ $ be the consecutive primes starting with
$\ p_0:=2.\ $ Let $\ M(n)\ $ be the multiplicative monomial
generated by $\ \{p_k:\ k=0\ldots n\}\ $ (of course $\ 1\in M(n)$).
Could you prove or ...
- 4,686
0
votes
0
answers
144
views
Remainder-balancedness of primes
Let $\mathbb{N}_+$ denote the set of positive integers. Consider the remainder function $\text{rem}:\mathbb{N}_+\times \mathbb{N}_+ \to \mathbb{N}\cup\{0\}$ defined by $$(n,d) \mapsto n - \Big(\Big\...
- 45.4k
-5
votes
1
answer
166
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 ...
- 397
2
votes
2
answers
249
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^*=\...
- 17.4k
2
votes
1
answer
124
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(...
- 2,949
14
votes
0
answers
292
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 $e$.
...
- 17.4k
2
votes
0
answers
67
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?
- 13.7k
1
vote
0
answers
62
views
Distribution of number of prime factors of $p^k\pm1$
What is the behavior of number of prime factors of integers of form $p^k\pm1$ where $p$ is a fixed odd prime or $2$ and $k$ varies over positive integers?
- 13.7k
9
votes
0
answers
190
views
primes concatenation sequence
Let us take a natural number x > 1. Then define a sequence $x_n$ as follows:
$x_0=x$;
if $x_n = p_1\cdots p_s$, where $p_1\leqslant\dots\leqslant p_s$ are prime numbers,
then $x_{n+1}$ is the ...
14
votes
1
answer
2k
views
Are there any Fibonacci numbers that are sandwiched between twin primes?
Note: These queries had come up during an earlier discussion: On Fibonacci numbers that are also highly composite. Am putting them up as a separate post.
Q: Are there any Fibonacci numbers that are ...
- 5,493
1
vote
1
answer
211
views
Lucas-Lehmer test for Wagstaff numbers?
Here is what I observed :
Let $N_p = 2^p+1$ and $W_p = (2^p+1)/3$ for Wagstaff numbers with $p$ a prime number > $3$.
Let the sequence $S_i = S_{i-1}^2 - 2$ with $S_0 = (2^{p-2}+1)/3$.
Then $W_p$ ...
- 119
13
votes
1
answer
570
views
A congruence for a product of binomial coefficients?
For every prime $p\geq 5$ one seems to have the congruence
$$(-1)^{(p-1)/2}\prod_{k=0}^{p-1}{p-1\choose k}\equiv 1-p+\frac{3}{2}p^2-\frac{7}{6}p^3\pmod{p^4}\ .$$
(I have checked all primes up to $5000$...
- 17.4k
2
votes
0
answers
229
views
Possible regularisation for sum of function of primes
Consider the following sum of function of primes:
$$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$
Here $p$ runs through all primes and $e$ is Euler's constant.
We can see that the sum ...
- 149
4
votes
1
answer
587
views
Reference for a proof of Euclid's Theorem for the infinitude of primes
I would be curious to have a reference for the following proof
of Euclid's Theorem on the infinitude of primes:
Using Legendre's formula (also called de Polignac's formula) for
$p$-adic valuations of ...
- 17.4k
3
votes
1
answer
870
views
What do we know about Lucky numbers?
I'm really fascinated by lucky numbers (Wikipedia; OEIS A000959) and their prime-like characteristics.
Wolfram states: write "out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The ...
- 39
6
votes
0
answers
219
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 ...
- 17.4k
12
votes
4
answers
2k
views
Proving Mertens' theorem using the prime number theorem
Mertens' Theorem states that
$$\sum_{p \leq x}\frac{1}{p} = \log \log x + M + O(1/\log x).$$
This is weaker than the prime number theorem; in fact according to the Wikipedia page, the prime number ...
- 20.8k
6
votes
1
answer
165
views
Is the set of all solutions $x > 0$ to $ \pi(x) = \operatorname{li}(x)$ unbounded?
Is the set of all solutions $x > 0$ to the equation $\pi(x) = \operatorname{li}(x)$ unbounded? Is $\liminf_{x \to \infty} |\pi(x)-\operatorname{li}(x)|$ equal to $0$?
Here, $\pi(x)$ denotes the ...
- 4,123