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
4
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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 ...
- 8,722
-1
votes
1
answer
164
views
An evaluation of the second Chebyshev function
Let
$$
\begin{align}
\Lambda (n) & &\text{the Von Mangoldt function,}\\
\psi(x)&:=\sum_{n=1}^{[x]}\Lambda (n)&\text{the econd Chebyshev function,}\\
T(x)&:=\sum_{n=1}^{[x]}\log(n). ...
- 99.1k
2
votes
2
answers
407
views
"Squeezing" the primes?
The logical idea here is to map a curve that encodes the primes into the region $(0,1)^2$ and analyze the distribution there more easily and achieve tight bounds.
To assess the distribution of primes, ...
- 1,061
4
votes
0
answers
157
views
On the asymptotic $\pi(x+h(x)) - \pi(x) \sim \frac{h(x)}{\log x} \ (x \to \infty)$
Let $h(x)$ be a function that is positive on $\mathbb{R}_{>0}$ and satisfies $h(x) = o(x)$ and $(\log x)^a = o(h(x))$ for all $a > 0$, as $x \to \infty$. Is it reasonable to expect under these ...
- 4,123
14
votes
1
answer
2k
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Are 0 and 1, respectively, the least and most used digits among primes?
In order to write the first 25 primes (2 to 97), 46 digits are necessary, nine of each of the digits 2, 3, and 7, fewer of all the others. Thereafter, at least for a while, the digit 1 is used more ...
- 3,679
13
votes
7
answers
4k
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Are there any interesting or lesser known proofs related to Bertrand's Postulate
There are 3 standard proofs of Bertrand's Postulate:
(1) Chebyshev's original proof
(2) Ramanujan's simplification of Chebyshev's proof
(3) Erdos's proof
I recently learned about the very ...
- 2,743
0
votes
0
answers
110
views
Question on the inverse Mellin transform $p(x)=\mathcal{M}_s^{-1}\left[-\xi(s)\,\frac{\zeta'(s)}{s\,\zeta(s)^2}\right]\left(\frac{1}{x}\right)$
Consider the function
$$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\, \...
- 1,061
20
votes
2
answers
2k
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Is every prime the largest prime factor in some prime gap?
Definition: In the gap between any two consecutive odd primes we have one or more composite numbers. One of these composite number will have a prime factor which is greater than that of any other ...
- 109k
2
votes
0
answers
101
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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 ...
- 595
8
votes
1
answer
1k
views
Unusual clump of small prime numbers?
\begin{align}
22097 & = 19\times1163 \\
22098 & = 2 \times 3 \times 29 \times 127 \\
22099 & = 7 \times 7 \times 11 \times 41 \\
22100 & = 2 \times2 \times 5 \times5 \times 13 \times ...
- 60.2k
2
votes
1
answer
695
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
$$f(x)=\sum\limits_{n=2}^x \frac{\Lambda(n)}{\...
- 11.9k
1
vote
2
answers
176
views
Prime factors bounded by $k$
Let $S$ be the set of integers with largest prime factor bounded by a given positive integer $k$. Is there a formula for the asymptotic density of such a set $S$?
- 11.9k
4
votes
0
answers
253
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A variant of the Green-Tao theorem
Green and Tao famously proved (The primes contain arbitrarily long arithmetic progressions) that there are arbitrarily long arithmetic progressions in the primes. Specifically, for $k = 3$ this ...
- 25.5k
2
votes
1
answer
98
views
Consecutive prime numbers in permutations of digits of the first consecutive positive integers
I have been toying for a while with the study of: in how many distinct primes and of which size can we divide permutations of digits of the first positive integers?
In this post I studied how many ...
- 1,119
12
votes
3
answers
3k
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111...11 base p = 111...11 base q
Feels like I am probably missing something obvious.
Are there distinct primes $p,q$ and positive integers $m,n$ such that
$$ \sum_{i=0}^{n} p^i = \sum_{j=0}^{m} q^j$$
Guessing the answer is no, but ...
CommunityBot
- 1
9
votes
2
answers
1k
views
On the error term of the Riemann explicit formula
Let: $\rho$ be a non-trivial zero of the Riemann zeta function, $\Lambda$ be the von-Mangoldt function and $\psi(x) =\sum_{n \leq x} \Lambda(n)$. What is the best known upper bound for
$$f(x, T) := \...
- 99.1k
4
votes
0
answers
252
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Asymptotic number of "modular primes"
We can say that a number $p$ is prime modulo $N$ if for any two numbers $1<a,b<p$, $ab \not\equiv p \pmod N$. We will define $p(n)$ to be the number of primes mod $n$. I'm wondering about the ...
- 2,979
1
vote
1
answer
341
views
On the number of primes between prime $p_n$ and $p_{n}^2$
does anyone knows if there are studies on the number of primes between prime $p_n$ and $p_{n}^2$, where $p_n$ is the $n$-th prime?
I am studying it through the following formula:
\begin{align}
\pi(p^...
- 2,578
1
vote
0
answers
88
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 $f(n)=0$ ...
- 11.9k
9
votes
2
answers
299
views
Representation of a residue modulo prime as a specific product
Let $p$ be a prime number. For every integer $m$, there are integers $u_1$, $u_2$, such that $\lvert u_1\rvert, \lvert u_2\rvert < \sqrt{p}$ and
$$m \equiv u_1u_2^{-1} \pmod{p}.$$
Proof of this ...
- 933
1
vote
0
answers
150
views
A transformation game for natural numbers?
Consider the completely additive function $\eta(n) := \sum_{p\mid n} v_p(n)p$ defined on natural numbers, with values in natural numbers. For literature, on this function, see the corresponding OEIS ...
24
votes
2
answers
4k
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Primes of the form $x^2+ny^2$ and congruences.
The answer of following classical problem is surely known, but I can't find a reference
For which positive integer $n$ is the set $S_n$ of primes of the form $x^2+n y^2$ ($x$, $y$ integers) ...
- 25.4k
1
vote
0
answers
458
views
Explicit formula for zeta function with special type of weight
Consider the following line of thinking:
$$\pi(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x} $$
Here,
$\operatorname{R}(x) = \...
- 690
3
votes
1
answer
238
views
Property of $3$-smooth numbers
Crossposted from math.stackexchange since there are no answers there.
Consider the sequence $a_k$ of $3$-smooth numbers (see OEIS A003586), i.e. the elements of:
$$S = \{ 2^i 3^j : i,j \ge 0 \}$$
in ...
- 25.5k
8
votes
0
answers
155
views
Hamiltonian paths in the prime sum graph
The following is a generalization of this old question .
Let $n\ge 2$, $[n]=\{1,\ldots,n\}$. For which distinct $a,b\in[n]$ is it possible to list $[n]$ in some order $x_1,\ldots,x_n$ such that $x_1=a$...
- 37.2k
2
votes
1
answer
924
views
A geometric proof that there are infinitely many primes?
Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{N})$.
Let $h(n) = J_2(n)$ be the second Jordan totient function, defined by:
$$J_2(n) = n^2 \prod_{p|n}(1-1/p^2)$$
...
- 2,462
0
votes
1
answer
223
views
Prime gap conjecture $ \pi_{2a}(n+(6a+4)^3)+(6a+4)^3 > \pi_{4a}(n)$ counterexamples?
Consider prime constellations $p,p+2s$ where both $p,p+2s$ are prime.
For instance for $s=1$ we get the twin primes.
We define the counting function $\pi_{2s}(n)$ to count the number of such pairs $p,...
- 8,984
21
votes
2
answers
2k
views
Implications of the disproof of the "climb-to-a-prime" conjecture
Now that James Davis has found a counter example, 13532385396179, to John Conway's climb-to-a-prime conjecture, I would be interested to learn whether this has any implications of interest in number ...
- 78.3k
0
votes
0
answers
90
views
Mathematical simplification of Willans' formula for the prime counting function
Willans' formula for the prime counting function $\pi (n)$ becomes computationally intractable for large values of $n$. Has there been any work specifically on reducing this computational complexity, ...
- 2,518
1
vote
0
answers
54
views
Largest interval containing family of sets with an overlap property
Here's a simplified version of a question I'm interested in.
Given $p$ and $q$ distinct prime numbers, we consider sets $A\subset \mathbb{N}\cup\{0\}, 0\in A$ of size $pq$, which are uniformly ...
- 539
8
votes
1
answer
895
views
Is this weak asymptotic Goldbach's conjecture open?
Let $\tau(x)$ be the number of even numbers $2<2n<x$ which can't be written as a sum of two primes.
Goldbach's conjecture: $\tau(x) = 0$
Asymptotic Goldbach's conjecture: $\tau(x) = O(1) $
...
- 99.1k
2
votes
1
answer
459
views
Number of points on a surface modulo p
I am guessing that the number of solutions $(x_1,x_2,\cdots ,x_s)$ modulo $p$ of the system of polynomials
$$x_1x_2\cdots x_s=1,$$
$$(x_1-1)(x_2-1)\cdots (x_s-1)=u$$
where $u$ is non-zero modulo $p$.
...
- 8,722
1
vote
0
answers
131
views
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\...
- 2,263
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votes
3
answers
2k
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For consecutive primes $a\lt b\lt c$, prove that $a+b\ge c$.
For consecutive primes $a\lt b\lt c$, prove that $a+b\ge c$.
I cannot find a counter-example to this. Do we know if this inequality is true? Alternatively, is this some documented problem (solved or ...
- 31.5k
1
vote
0
answers
144
views
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 $...
- 11.9k
4
votes
0
answers
304
views
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
0
votes
0
answers
148
views
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 ...
- 13.7k
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0
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95
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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
5
votes
0
answers
160
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Reducibility of $f(x)^{2^n}+1$ and $f(x)^{2^n}+g(x)^{2^n}$
Related to generalized Fermat numbers.
Let $f(x),g(x)$ be coprime polynomials with integer coefficients.
Assume that if $f(x)$ or $g(x)$ are of the form $h(x)^k$ then $k$ is power
of two.
Q1 Is it ...
- 24.2k
-2
votes
2
answers
149
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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 $1$.
- 30.6k
3
votes
1
answer
214
views
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 ...
- 20.8k
2
votes
1
answer
156
views
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 ...
- 20.8k
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
32
votes
1
answer
769
views
Can we write each positive rational number as $\frac1{p_1-1}+\ldots+\frac1{p_k-1}$ with $p_1,\ldots,p_k$ distinct primes?
It is well-known that any positive rational number can be written as the sum of finitely many distinct unit fractions. This is easy since
$$\frac1n=\frac1{n+1}+\frac1{n(n+1)}\quad\text{for all}\ n=1,2,...
- 6,608
2
votes
1
answer
136
views
Estimating the minimum number of distinct least prime factors found in range of $c$ consecutive integers
When I look at the count of distinct least prime factors for a range of consecutive integers, I am seeing the same minimum number appear again and again. I am wondering if this number represents the ...
- 1,039
12
votes
1
answer
781
views
Quadratic reciprocity for three primes?
The quadratic reciprocity law states that for $p_1\ne p_2$ prime, the product $\left(\frac{p_1}{p_2}\right)\left(\frac{p_2}{p_1}\right)$ takes values $1$ or $-1$ depending on whether $p_1$ and $p_2$ ...
- 44.7k
3
votes
1
answer
366
views
What heuristic arguments support Montgomery's conjecture for primes in short intervals?
I have a question regarding a conjecture due to H. L. Montgomery on the number of primes in short intervals. The conjecture apparently arises from probabilistic reasoning upon assuming the Riemann ...
- 405
11
votes
1
answer
2k
views
Distribution of the number of prime factors
Count the number of prime factors of a number $n$
to include multiplicity,
so that
$$n=24=2^3 \cdot 3 = 2 \cdot 2 \cdot 2 \cdot 3$$
has $4$ prime factors, and
$$n =
6500 =
2^2 \cdot 5^3 \cdot 13 =
2 \...
12
votes
1
answer
821
views
Numbers $k$ with $\{\binom nk:\ n\in\mathbb N\}$ dense in $\mathbb Z_p$ for any prime $p\le k$
Let $k$ be a positive integer and let $p$ be a prime. In my 2011 PAMS paper joint with my former student W. Zhang [Proc. Amer. Math. Soc. 139(2011), 1569-1577], we studied when $$S(k)=\left\{\binom nk:...
CommunityBot
- 1
5
votes
2
answers
344
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