All Questions
Tagged with nt.number-theory prime-numbers
1,808 questions
11
votes
1
answer
324
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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 $...
- 5,470
0
votes
1
answer
292
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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 $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , ...
- 3,319
1
vote
0
answers
72
views
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?:
$$
\prod_{p \in A} p \ + \ \prod_{p \in B} p \ = ...
- 19.6k
3
votes
1
answer
747
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Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?
Is
$$1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n},$$
where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?
Context:
This question came out as a result in ...
CommunityBot
- 1
6
votes
1
answer
1k
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A question about the Möbius Function
$\newcommand\bmod{\mathbin\%}$I have been playing around with the Möbius Function and primorials and I am finding results that I am not yet able to understand which I suspect are very elementary.
Here'...
- 12.9k
5
votes
1
answer
737
views
Smallest prime factor of numbers
The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\...
- 105k
7
votes
1
answer
493
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About semiprimal representations of $1$
Conjecture $A_1$ : For every $m \in \mathbb N \setminus \{1\}$ there exist mutually different primes $p_{r_1},...,p_{r_m}$ and mutually different primes $b_{w_1},...,b_{w_m}$ and numbers $i_1,...,i_m \...
CommunityBot
- 1
1
vote
1
answer
253
views
Prime divisors of $p^n-1$, primitive prime divisors
Let $p,q,t_1,t_2$ be distinct prime numbers and let
$$k=\frac{p^{qt_1t_2}-1}{p^q-1}.$$
Suppose that $\gcd(k,qt_1t_2)=1$. Is there any reason that $k$ is divisible by at least $7$ distinct prime ...
- 22.5k
1
vote
2
answers
390
views
Solving a recurrence relation for the prime counting function?
I have found some number sequence $c_n = 1+b_n$ for $n \ge 0$, where $b_n = $ A307977(n).
I am trying to solve the following recurrence relation for the prime counting function:
$$\forall n \ge 3: \pi(...
- 1,126
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 ...
- 5,089
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 ...
- 14.6k
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
7
votes
0
answers
335
views
Residues of consecutive primes modulo a fixed integer
It is well-known that the primes are uniformly distributed in residue classes modulo any fixed integer. More precisely, for each integer $q$ and each residue $a \in \mathbb{Z}/q\mathbb{Z}$ that is ...
CommunityBot
- 1
4
votes
1
answer
559
views
Goldbach for certain classes of $n$
Asked on MSE without response here.
$\#$ of ways even $n$ can be represented by prime additions is hereafter denoted $G(n)$.
The Wiki article on the Goldbach conjecture states that
In 1975, Hugh ...
- 39.9k
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 $n=p_{1}p_{2}p_{3}\dotsb p_{k}$ ...
- 109k
3
votes
1
answer
348
views
On conjectures about the arithmetic function that counts the number of Sophie Germain primes
I've edited this post two years ago on Mathematics Stack Exchange, with identifier 3590406 and same title On conjectures about the arithmetic function that counts the number of Sophie Germain primes, ...
- 148k
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
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 ...
- 4,713
6
votes
1
answer
669
views
Recursivity of the primes
As is well-known, with the finite set of the first $n$ primes $p_1,\dots ,p_n$ one can find all primes exactly (i.e. without false positives) with the Eratosthenes Sieve in the interval $[p_n+1;p_{n+1}...
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
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
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
39
votes
1
answer
2k
views
Prime number races in 2 dimensions
Is the mapping $$f: \ \mathbb{N} \rightarrow \mathbb{Z}[i], \ \ \ n \ \mapsto
\sum_{2 < p \leq n \ {\rm prime}} e^{\frac{p-1}{4} \pi i}$$ surjective?
In 1999, when I was an undergraduate student, ...
- 2,821
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 $n\in\mathbb{N}$?
- 39.9k
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?
- 63.9k
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 $v(i)=m(i+1)-2$...
CommunityBot
- 1
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\...
- 16.2k
6
votes
2
answers
723
views
Does always exist a prime number between $n(n+1)$ and $(n+1)(n+2)$?
Let $n$ is positive integer number, does always exist a prime number between $n(n+1)$ and $(n+1)(n+2)$?
- 433
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
25
votes
7
answers
3k
views
Question on consecutive integers with similar prime factorizations
Suppose that $n=\prod_{i=1}^{k} p_i^{e_i}$ and $m=\prod_{i=1}^{l} q_i^{f_i}$ are prime factorizations of two positive integers $n$ and $m$, with the primes permuted so that $e_1 \le e_2 \cdots \le e_k$...
- 4,713
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, ...
- 6,198
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 ...
- 1,028
3
votes
1
answer
288
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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 $n>100$?
- 34.3k
5
votes
2
answers
388
views
How much do these interval collections cover?
As usual any related references are appreciated.
Let $p \lt q$ be distinct primes, and for all such pairs, let $m=pq$ and let $\cal{C}$ be the collection $(m-p,m)$ of open intervals. Does (the union ...
- 34.3k
3
votes
1
answer
2k
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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 ...
- 10.4k
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 ...
- 8,842
-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). ...
- 105k
2
votes
2
answers
424
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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, ...
- 1,126
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
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 ...
- 2,821
20
votes
2
answers
2k
views
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 ...
- 114k
2
votes
0
answers
103
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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 ...
- 869
8
votes
1
answer
1k
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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 ...
- 63.9k
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
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 $S$?
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
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 ...
- 1,451
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
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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) := \...
- 105k
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