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.
2,002
questions
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160
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Is the integer factorization into prime numbers normally distributed?
Let $P_1(n) := 1$ if $n=1$ and $\max_{q|n, \text{ }q\text{ prime}} q$ otherwise, denote the largest prime divisor of $n$.
Let us define some rooted trees $T_{n,m}$ for $1 \le m \le n$ by:
$T_{n,m}$ ...
- 2,126
4
votes
0
answers
107
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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 $N_P$ ...
- 41
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0
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222
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Is parametric formula for congruent numbers worth publishing? [closed]
Earlier in my undergrad career (around 2020) I discovered that all congruent numbers are divisors of Pythagorean triples. Specifically, I can show that they can be written as $$\frac{nm\left(m-n\right)...
- 13
5
votes
2
answers
391
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On the number of distinct prime factors of $p^2+p+1$
Is it true that, for each positive integer $c$, there exists a prime number $p$ such that $p^2+p+1$ is divisible by at least $c$ distinct primes?
- 886
8
votes
1
answer
580
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Arithmetic sequences and Artin's conjecture
(Sorry if this is a naive question; it is not my area!)
Consider the following strengthening of Artin's conjecture on primitive roots (and Dirichlet's theorem) for the case of $n=2$: every arithmetic ...
0
votes
1
answer
238
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Factorization trees and (continued) fractions?
This question is inspired by trying to understand the lexicographic sorting of the natural numbers in the fractal at this question:
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , ...
- 2,126
1
vote
0
answers
64
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Is there an upper bound on the number of partitions of a finite set of primes into 3 sets the products of 2 of which sum to the product of the third?
Is there an upper bound on the number of partitions of a finite set $S$
of prime numbers into 3 sets $A$, $B$ and $C$ for which the following holds?:
$$
\prod_{p \in A} p \ + \ \prod_{p \in B} p \ = ...
- 19.3k
4
votes
1
answer
408
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Smallest prime factor of numbers
The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\...
- 191
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294
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Does the mean ratio of the largest prime factor in prime gaps to the lower bound of the gap converge?
Posting in MO since this questions has been unanswered in MSE for 3 months.
Let $p_n$ be the $n$-th prime and $q_n$ be largest among all the prime factors of the composite numbers between $p_n$ and $...
- 4,628
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1
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682
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Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?
Is
$$1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n},$$
where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?
Context:
This question came out as a result in ...
- 2,126
-1
votes
0
answers
160
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Size and type of a Galois radius: are they correlated?
Say as usual that a non negative integer $r$ is a Galois radius of $n$ of type $(a,b)$ if $(n-r,n+r)=(p^a,q^b)$ for some couple of primes $(p,q)$. As primes tend to get sparser as we progress along ...
- 6,954
4
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0
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368
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How to define a fractal from the lexicographic sorting on the prime factorization of natural numbers?
Consider on the natural number the lexicographic ordering on the prime factorization:
If $m = p_1^{a_1}\cdots p_r^{a_r},n = q_1^{b_1}\cdots q_s^{b_s}$ then we define:
$$m \vartriangleleft n :\iff [(...
- 2,126
1
vote
1
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198
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Prime divisors of $p^n-1$, primitive prime divisors
Let $p,q,t_1,t_2$ be distinct prime numbers and let
$$k=\frac{p^{qt_1t_2}-1}{p^q-1}.$$
Suppose that $\gcd(k,qt_1t_2)=1$. Is there any reason that $k$ is divisible by at least $7$ distinct prime ...
- 886
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1
answer
192
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A question about the prime counting function
I was playing around with the prime counting function and came across something that seemed correct to me, maybe it's already been proven but I don't know so I decided to ask here.
maybe a stupid ...
- 61
1
vote
2
answers
360
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Solving a recurrence relation for the prime counting function?
I have found some number sequence $c_n = 1+b_n$ for $n \ge 0$, where $b_n = $ A307977(n).
I am trying to solve the following recurrence relation for the prime counting function:
$$\forall n \ge 3: \pi(...
- 2,126
7
votes
2
answers
2k
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Can every integer be written as a sum of squares of primes?
This question is mainly inspired from a different problem I was working on.
Is there a value of $k$ such that, for each $n\in \mathbb N$, the equation
$$\sum_{i=1}^{k}x_i^2=n$$
is solvable in $x_1,\...
- 661
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votes
1
answer
233
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Density of primes $p$ where $p-1$ has a prime factor exceeding $p^{2/3}$
Fouvry proved* that primes $p$ such that the greatest prime factor, $q$, of $p-1$ is greater than $p^{2/3}$ have positive density in the primes. (The sequence is A073024 in the OEIS.)
Are there any ...
- 8,914
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0
answers
124
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Limit of scaled infinite sum with Dirichlet characters modulo 4: is it zero?
I am trying to get an asymptotic formula such as
$$ L_4(s, n) \sim L_4(s) + \rho_n(s)\Lambda_n + \frac{\alpha(s)}{\sqrt{n}} + \frac{\beta(s)}{\sqrt{n\log n}}+\cdots$$
where $L_4(s, n)$ is the first $n$...
- 3,058
3
votes
2
answers
385
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If $p_1$ and $p_2$ are prime numbers, then either $p_1$ divides $\sum_{i=1}^{p_1-1} i^{p_1p_2-1}$ or $p_2$ divides $\sum_{i=1}^{p_2-1} i^{p_1p_2-1}$?
I feel like it's true as for small cases I couldn't find counterexample.
In general, whether it's true that if we have prime number, $p_{1}, p_{2},\dotsc, p_{k}$ and $n=p_{1}p_{2}p_{3}\dotsb p_{k}$ ...
2
votes
0
answers
112
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On the integer of the form p^a q^b closest to a given integer N
If we give ourselves a number having only one prime factor $p$ and a given natural integer $N$, we know how to give the integer of the form $p^k$ closest (and less than) to this integer $N$ it's ...
- 21
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votes
2
answers
1k
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Prime gaps within which every "small" prime appears as a factor: Are there only finitely many? Is this the last one?
For a bounded range of positive integers $n,n+1,\ldots,m,$ call a prime number "small" if it does not exceed $\sqrt m,$ so that if one is trying to factor all of these numbers into primes, ...
- 11.4k
2
votes
1
answer
210
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Prime gaps that are "relatively" bigger than all later prime gaps: Is this in OEIS?
This OEIS entry is about
Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k.
I'm wondering about a different ...
- 11.4k
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0
answers
227
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The number of continuously increasing primes gaps in the interval $[2,n]$ is less than $\log n$
A prime gap is the difference between two successive prime numbers. The $n$-th prime gap, denoted gn or $g(p_n)$ is the difference between the $(n+1)$-st and the $n$-th prime numbers. Using my ...
- 1,076
4
votes
1
answer
229
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Irreducible integral polynomials having roots module primes in arithmetic progressions
Let $f(x)$ be an irreducible polynomial with integer coefficients. One can show (see Exercise 7.2 in this paper of Lenstra) that if $f(x)=0$ has a solution mod $p$ for all but finitely many primes $p$...
- 6,014
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0
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264
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Can you prove and/or generalize this formula involving the Möbius function at n = square free numbers for elliptic curve related sequence in the OEIS?
Let $g(n)$ be the Dirichlet inverse of the Euler totient function:
$$g(n) = \sum\limits_{d|n} d \cdot \mu(d)$$
and let $f(x,y)$ be the elliptic equation:
$$f(x,y)=x^3 - x^2 - y^2 - y$$
Show that the ...
- 1,133
8
votes
2
answers
756
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Natural density of the set of simple numbers
Let us call $n>1$ simple if every prime power $q$ with $q-1 \mid n-1$ is a prime number. (Please let me know if there is already an established name for these numbers.) The simple numbers $\leq 100$...
13
votes
1
answer
640
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When is $\mathrm{gcd}(k,p^k-1)=1$ true?
Let $p$ be a prime. Is there a classification of the numbers $k \geq 1$ such that $\gcd(k,p^k-1)=1$? If not, can we at least produce an explicit infinite subset? What is known about these $k$?
For the ...
0
votes
1
answer
174
views
Primes above the distant prime neighbors
Let $\ \mathbb P\ $ be the set of all natural primes. Pair $\ (p\ q)\ $ are prime
neighbors $\ \Leftarrow:\Rightarrow$
$$ \{x\in\mathbb Z: p\le x\le q\}\cap\mathbb P\,\ =\,\ \{p\,\ q\} $$
Prime $\ x\...
- 4,674
3
votes
2
answers
395
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Approximation of partial sum over prime omega function
I asked the question in Math StackExchange. Link: https://math.stackexchange.com/questions/4765476/approximation-of-partial-sum-over-prime-omega-function
I haven't got any response yet. Here are the ...
- 221
2
votes
1
answer
272
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Factorizations of cyclotomic polynomials valuated at primes
I have a question concerning cyclotomic polynomials valuated at primes. I first state it in the easiest possible form.
There exists a function $f:\mathbb{N}\to\mathbb{N}$ such that, if $p$ is a prime, ...
- 886
2
votes
0
answers
199
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Not a twin prime pair test using $\gcd$ only
Let $m$ be an odd positive integer such that $m=2k+1$, $k\in\mathbb{N}$.
Let $v$ be a vector of $n$ positive integers. Let $v(i)$ be the $i$-th element of the vector. Then we start with $v(i)=m(i+1)-2$...
- 3,018
1
vote
1
answer
301
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Inequality of three prime factors of $x^2-1$
This is about my experimental math observation on prime factorization of $x^2-1$
We can see, for many $x\in \mathbb{Z}_+$, the expression $x^2-1=(x-1)(x+1)$ gives result as a product of twin prime. ...
- 537
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3
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What is the simplest proof that the density of coprime pairs does not go to zero?
By density of coprime pairs, I mean the proportion of pairs integers between $1$ and $x$ which are coprime.
This is known to be asymptotically $1/\zeta(2)$.
I want something much weaker, namely that ...
- 18.7k
3
votes
1
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277
views
For any integer $n>6$, does there always exist a prime $p>n+1$ such that $p\mid 2^n-1$?
For any integer $n>6$, does there always exist a prime $p>n+1$ such that $p\mid 2^n-1$?
It's true for $6<n<100$. But for $n>100$?
- 103
4
votes
1
answer
247
views
For any integer $n>0$, does there always exist a prime $p>n$ such that $p\mid 2^n-1$?
For any integer $n>0$, does there always exist a prime $p>n$ such that $p\mid 2^n-1$?
It's easy to verify this result for $1<n<100$ by computer.
But for any integer $n>0$, is it always ...
- 103
1
vote
1
answer
295
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Some necessary condition for $\gcd(m,n) $ be a proper divisor of $\gcd(mk_2 +nk_1,mn) $ [closed]
Let $m,n,k_1,k_2 $ be natural numbers such that $(k_1,m)=(k_2,n)=1 $.
Statement 1: $\gcd(m,n) $ is a proper divisor of $\gcd(mk_2 +nk_1,mn) $, for every $k_1,k_2$ having the above property.
Statement ...
- 913
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votes
2
answers
387
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"Squeezing" the primes?
The logical idea here is to map a curve that encodes the primes into the region $(0,1)^2$ and analyze the distribution there more easily and achieve tight bounds.
To assess the distribution of primes, ...
- 141
4
votes
0
answers
154
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On the asymptotic $\pi(x+h(x)) - \pi(x) \sim \frac{h(x)}{\log x} \ (x \to \infty)$
Let $h(x)$ be a function that is positive on $\mathbb{R}_{>0}$ and satisfies $h(x) = o(x)$ and $(\log x)^a = o(h(x))$ for all $a > 0$, as $x \to \infty$. Is it reasonable to expect under these ...
- 4,113
0
votes
0
answers
92
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Question on the inverse Mellin transform $p(x)=\mathcal{M}_s^{-1}\left[-\xi(s)\,\frac{\zeta'(s)}{s\,\zeta(s)^2}\right]\left(\frac{1}{x}\right)$
Consider the function
$$p(x)=\underset{K\to\infty}{\text{lim}}\left(\sum\limits_{k=1}^K \Lambda(k) \left(\frac{2 \pi k^2}{x^2}-1\right) e^{-\frac{\pi k^2}{x^2}}\right)\tag{1}$$
where
$$P(s)=s\, \...
- 1,001
-1
votes
1
answer
160
views
An evaluation of the second Chebyshev function
Let
$$
\begin{align}
\Lambda (n) & &\text{the Von Mangoldt function,}\\
\psi(x)&:=\sum_{n=1}^{[x]}\Lambda (n)&\text{the econd Chebyshev function,}\\
T(x)&:=\sum_{n=1}^{[x]}\log(n). ...
- 417
14
votes
1
answer
2k
views
Are 0 and 1, respectively, the least and most used digits among primes?
In order to write the first 25 primes (2 to 97), 46 digits are necessary, nine of each of the digits 2, 3, and 7, fewer of all the others. Thereafter, at least for a while, the digit 1 is used more ...
- 3,497
2
votes
0
answers
98
views
On equidistribution of primes in positive characteristic
In S. Lang's book "Algebraic Number Theory" (1986), page 317, Theorem 6 states essentially that given $P$ a set of primes, let $\tau:P\longrightarrow J$ be the typical idèle map taking ...
- 655
2
votes
1
answer
684
views
Does the Riemann hypothesis predict a bound for this prime-counting function?
Does the Riemann hypothesis predict an upper bound for
$$\left|f(x)-\left(\operatorname{li}(x)-\frac{x}{\log x} \right)\right|,\quad x\ge 2\tag{1}$$
where
$$f(x)=\sum\limits_{n=2}^x \frac{\Lambda(n)}{\...
- 1,001
4
votes
0
answers
243
views
A variant of the Green-Tao theorem
Green and Tao famously proved (The primes contain arbitrarily long arithmetic progressions) that there are arbitrarily long arithmetic progressions in the primes. Specifically, for $k = 3$ this ...
- 25k
1
vote
2
answers
174
views
Prime factors bounded by $k$
Let $S$ be the set of integers with largest prime factor bounded by a given positive integer $k$. Is there a formula for the asymptotic density of such a set $S$?
- 397
2
votes
1
answer
87
views
Consecutive prime numbers in permutations of digits of the first consecutive positive integers
I have been toying for a while with the study of: in how many distinct primes and of which size can we divide permutations of digits of the first positive integers?
In this post I studied how many ...
- 289
9
votes
2
answers
1k
views
On the error term of the Riemann explicit formula
Let: $\rho$ be a non-trivial zero of the Riemann zeta function, $\Lambda$ be the von-Mangoldt function and $\psi(x) =\sum_{n \leq x} \Lambda(n)$. What is the best known upper bound for
$$f(x, T) := \...
- 120
12
votes
3
answers
3k
views
111...11 base p = 111...11 base q
Feels like I am probably missing something obvious.
Are there distinct primes $p,q$ and positive integers $m,n$ such that
$$ \sum_{i=0}^{n} p^i = \sum_{j=0}^{m} q^j$$
Guessing the answer is no, but ...
- 129
19
votes
2
answers
1k
views
Does this number exist?
Does there exist $x\in\mathbb{R}$ such that $\lfloor 10^nx\rfloor$ is a prime number for all $n\in\mathbb{N}$?
- 3,609
1
vote
0
answers
85
views
Validity of analysis of summation of function of primes using Abel–Plana summation:
Consider the analytic function $g(x)$
Define
$$f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$$
Note that:
$$f(p)=g(p) \text{ for prime } p$$
And $f(n)=0$ ...
- 690