All Questions
Tagged with nt.number-theory prime-numbers
1,808 questions
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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
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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
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106
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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
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2
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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 $1$.
- 577
2
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1
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775
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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
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2
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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
93
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3
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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 ...
- 861
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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":
$$
\sum_{p \leq n} \frac{\log p}{p -...
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$p^2+a^2$ can be a squarefree number with all prime divisors less than $p$?
Let $p$ be a prime $\ge 31$.
Is there an integer $a < p$ such that $p^2 + a^2$ is a squarefree and all of its prime divisors are less than $p$?
For example, for $p=31$, $31^2+5^2 = 986 = 2 \times ...
- 483
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151
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Mertens' Third Theorem for primes of the form $4n+1$
I am looking for upper and lower bounds for the following expression:
$$\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 \...
- 193
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340
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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 ...
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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 ...
- 5,470
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101
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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
...
- 17.9k
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151
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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 ...
3
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328
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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 ...
- 387
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687
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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
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234
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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 ...
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289
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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 ...
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515
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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 ...
- 309
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125
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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\}...
- 34.3k
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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
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481
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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} - (\...
- 521
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1
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262
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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?
- 356
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567
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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 ...
- 13.3k
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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 ...
1
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179
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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$.
...
- 25.4k
1
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151
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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
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107
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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 ...
4
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1
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389
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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
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321
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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 ...
- 4,786
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146
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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\...
- 48.9k
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260
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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^*=\...
- 17.9k
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140
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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(...
- 3,062
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297
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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 $e$.
...
- 17.9k
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70
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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?
- 13.9k
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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?
- 13.9k
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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 ...
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1
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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 ...
- 5,979
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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 $W_p$ ...
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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 $5000$...
- 17.9k
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238
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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
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1
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601
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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 ...
- 17.9k
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 ...
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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 ...
- 17.9k
13
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4
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2k
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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 ...
- 21.3k
4
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1
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251
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Density of semiprimes in arithmetic progression
Let $n,a,b$ be integers such that $n$ and $a$ are coprime, and $n$ and $b$ are also coprime. According to the Prime number theorem for arithmetic progressions, the primes which are $a\mod n$ have the ...
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Prime numbers and number of partitions of $n$ into distinct parts with boundary size $2$
Let $a(n)$ be A227559, i.e., number of partitions of $n$ into distinct parts with boundary size $2$. Be careful here: offset is $3$.
I conjecture that $a(4n+2)=2n+1$ for $n>0$ if and only if $2n+1$ ...
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3
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Is there a $c > 1$ such that for all $n \ge 1$ the largest integer $\le c^n$ is prime?
Does there exist a real number $c > 1$ such that for every natural number $n > 0$, the number $\lfloor c^n \rfloor$ is prime?
I doubt such a number $c$ is known to exist, since the best similar ...
- 22.3k
6
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2
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575
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Average value of the prime omega function $\Omega$ on predecessors of prime powers
For a positive integer $n$, the prime omega function value $\Omega(n):=\sum_{p\mid n}{\nu_p(n)}$ counts the number of prime divisors of $n$ with multiplicities. A result of Hardy and Wright, [1, ...
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9
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Why is integer factoring hard while determining whether an integer is prime easy?
In 2002, the discovery of the AKS algorithm proved that it is possible to determine whether an integer is prime in polynomial time deterministically. However, it is still not known whether there is an ...
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