All Questions
Tagged with nt.number-theory prime-numbers
1,808 questions
11
votes
1
answer
324
views
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 $...
- 5,470
3
votes
1
answer
747
views
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 ...
- 3,064
6
votes
1
answer
479
views
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:
$$m \vartriangleleft n :\iff [(...
- 3,064
1
vote
1
answer
253
views
Prime divisors of $p^n-1$, primitive prime divisors
Let $p,q,t_1,t_2$ be distinct prime numbers and let
$$k=\frac{p^{qt_1t_2}-1}{p^q-1}.$$
Suppose that $\gcd(k,qt_1t_2)=1$. Is there any reason that $k$ is divisible by at least $7$ distinct prime ...
- 934
0
votes
1
answer
264
views
A question about the prime counting function
I was playing around with the prime counting function and came across something that seemed correct to me, maybe it's already been proven but I don't know so I decided to ask here.
maybe a stupid ...
- 61
1
vote
2
answers
390
views
Solving a recurrence relation for the prime counting function?
I have found some number sequence $c_n = 1+b_n$ for $n \ge 0$, where $b_n = $ A307977(n).
I am trying to solve the following recurrence relation for the prime counting function:
$$\forall n \ge 3: \pi(...
- 3,064
10
votes
2
answers
3k
views
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,\...
- 791
4
votes
1
answer
286
views
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 ...
- 9,114
2
votes
0
answers
131
views
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 $n$...
- 3,098
3
votes
2
answers
409
views
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
120
views
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 ...
- 69
4
votes
0
answers
335
views
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 $g_n$ or $g(p_n)$ is the difference between the $(n+1)$-st and the $n$-th prime numbers. Using my ...
- 1,776
3
votes
0
answers
330
views
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 ...
- 1,183
8
votes
2
answers
1k
views
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 $\leq 100$...
- 63.1k
13
votes
1
answer
700
views
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 ...
- 63.1k
0
votes
1
answer
177
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\...
- 4,786
3
votes
2
answers
580
views
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
3
votes
1
answer
421
views
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, ...
- 934
2
votes
0
answers
199
views
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 $v(i)=m(i+1)-2$...
- 4,938
20
votes
3
answers
3k
views
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 ...
- 19k
3
votes
1
answer
288
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
259
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
594
views
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 ...
- 923
2
votes
2
answers
424
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, ...
- 383
4
votes
0
answers
160
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 ...
- 4,985
-1
votes
1
answer
170
views
An evaluation of the second Chebyshev function
Let
$$
\begin{align}
\Lambda (n) & &\text{the Von Mangoldt function,}\\
\psi(x)&:=\sum_{n=1}^{[x]}\Lambda (n)&\text{the econd Chebyshev function,}\\
T(x)&:=\sum_{n=1}^{[x]}\log(n). ...
- 328
16
votes
3
answers
2k
views
Are 0 and 1, respectively, the least and most used digits among primes?
In order to write the first 25 primes (2 to 97), 46 digits are necessary, nine of each of the digits 2, 3, and 7, fewer of all the others. Thereafter, at least for a while, the digit 1 is used more ...
- 3,707
2
votes
0
answers
103
views
On equidistribution of primes in positive characteristic
In S. Lang's book "Algebraic Number Theory" (1986), page 317, Theorem 6 states essentially that given $P$ a set of primes, let $\tau:P\longrightarrow J$ be the typical idèle map taking ...
- 675
2
votes
1
answer
740
views
Does the Riemann hypothesis predict a bound for this prime-counting function?
Does the Riemann hypothesis predict an upper bound for
$$\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)}{\...
- 1,126
4
votes
0
answers
267
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 ...
- 26.9k
1
vote
2
answers
182
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
106
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 ...
- 189
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
$$f(x, T) := \...
- 1,019
12
votes
3
answers
3k
views
111...11 base p = 111...11 base q
Feels like I am probably missing something obvious.
Are there distinct primes $p,q$ and positive integers $m,n$ such that
$$ \sum_{i=0}^{n} p^i = \sum_{j=0}^{m} q^j$$
Guessing the answer is no, but ...
- 129
19
votes
2
answers
1k
views
Does this number exist?
Does there exist $x\in\mathbb{R}$ such that $\lfloor 10^nx\rfloor$ is a prime number for all $n\in\mathbb{N}$?
- 4,074
1
vote
0
answers
104
views
Validity of analysis of summation of function of primes using Abel–Plana summation:
Consider the analytic function $g(x)$
Define
$$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$$
And $f(n)=0$ ...
- 790
8
votes
1
answer
1k
views
Unusual clump of small prime numbers?
\begin{align}
22097 & = 19\times1163 \\
22098 & = 2 \times 3 \times 29 \times 127 \\
22099 & = 7 \times 7 \times 11 \times 41 \\
22100 & = 2 \times2 \times 5 \times5 \times 13 \times ...
4
votes
0
answers
262
views
Asymptotic number of "modular primes"
We can say that a number $p$ is prime modulo $N$ if for any two numbers $1<a,b<p$, $ab \not\equiv p \pmod N$. We will define $p(n)$ to be the number of primes mod $n$. I'm wondering about the ...
- 3,319
1
vote
0
answers
153
views
A transformation game for natural numbers?
Consider the completely additive function $\eta(n) := \sum_{p\mid n} v_p(n)p$ defined on natural numbers, with values in natural numbers. For literature, on this function, see the corresponding OEIS ...
- 3,064
9
votes
2
answers
319
views
Representation of a residue modulo prime as a specific product
Let $p$ be a prime number. For every integer $m$, there are integers $u_1$, $u_2$, such that $\lvert u_1\rvert, \lvert u_2\rvert < \sqrt{p}$ and
$$m \equiv u_1u_2^{-1} \pmod{p}.$$
Proof of this ...
- 965
0
votes
0
answers
352
views
On a Duality between Riemann-weil explicit formula and Abel- Plana summation of trigonometric prime counting function:
Consider the analytic function $g(x)$
Now define
$f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$
Such that
$|f(x+it)|=o(e^{2πt})$
uniformly for every $x$...
- 790
3
votes
1
answer
267
views
Property of $3$-smooth numbers
Crossposted from math.stackexchange since there are no answers there.
Consider the sequence $a_k$ of $3$-smooth numbers (see OEIS A003586), i.e. the elements of:
$$S = \{ 2^i 3^j : i,j \ge 0 \}$$
in ...
- 1,000
8
votes
0
answers
164
views
Hamiltonian paths in the prime sum graph
The following is a generalization of this old question .
Let $n\ge 2$, $[n]=\{1,\ldots,n\}$. For which distinct $a,b\in[n]$ is it possible to list $[n]$ in some order $x_1,\ldots,x_n$ such that $x_1=a$...
- 37.7k
8
votes
1
answer
205
views
Are there infinite numbers of the form $\sigma_1(n)=\sigma_1(m)=p$, or is there only one?
I put forward a hypothesis in number theory, it is as follows.$ \sigma_1(n)=\sigma_1(m)=p$, where $\sigma_1$ is the divisor sum function, $n,m\in \mathbb N$, and $p$ is prime. I recently noticed and ...
- 125
3
votes
1
answer
954
views
A geometric proof that there are infinitely many primes?
Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{N})$.
Let $h(n) = J_2(n)$ be the second Jordan totient function, defined by:
$$J_2(n) = n^2 \prod_{p|n}(1-1/p^2)$$
...
- 3,064
1
vote
0
answers
55
views
Largest interval containing family of sets with an overlap property
Here's a simplified version of a question I'm interested in.
Given $p$ and $q$ distinct prime numbers, we consider sets $A\subset \mathbb{N}\cup\{0\}, 0\in A$ of size $pq$, which are uniformly ...
- 549
0
votes
1
answer
241
views
Prime gap conjecture $ \pi_{2a}(n+(6a+4)^3)+(6a+4)^3 > \pi_{4a}(n)$ counterexamples?
Consider prime constellations $p,p+2s$ where both $p,p+2s$ are prime.
For instance for $s=1$ we get the twin primes.
We define the counting function $\pi_{2s}(n)$ to count the number of such pairs $p,...
- 763
2
votes
3
answers
539
views
When does the sum of squares of initial primes equal a triangular number?
Let $(p_i)$ be the sequence of prime numbers. Can we solve the equation:
$$\sum_{i=1}^k p_i^2=\frac{n(n+1)}{2}$$
in $(k,n)$? Note that $(7,36)$ is solution. Is this the unique solution?
- 377
3
votes
2
answers
802
views
Goldbach conjecture and the difference of two primes
The Goldbach conjecure is not yet proved. But, when an even number is represented as a sum of two primes, is there any knwon result about the difference of the two primes?
That is, if $2n$ is a sum of ...
- 483
1
vote
0
answers
132
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\...
- 2,605