Tagged Questions
10
votes
3answers
71 views
$\sum\limits_{\text{prime }p} 2^{-p}$ is an irrational number
I need help to prove the following result.
$\displaystyle\sum_{\text{prime }p} 2^{-p}$ is an irrational number.
3
votes
2answers
32 views
Right triangles with integer sides
Most of you know these triples:
$3: 4 :5$
$5: 12 :13$
$8: 15 :17$
$7: 24 :25$
$9: 40 :41$
More generally we can construct such triangles such as
$$2x:x^2-1:x^2+1$$
My question is why one of ...
3
votes
0answers
57 views
The divergence of the series of reciprocals of primes (proof check):
I just wanted to check my attempt at a proof for the divergence of:
$$\sum_{n=1}^{\infty} \frac{1}{p_n} \tag{ $\star$ }$$
We begin with assuming that $(\star)$ converges. If $(\star)$ ...
4
votes
1answer
59 views
$p^{3}+m^{2}$ is square of a number.
Well i thought it is a nice problem so i will post it here.
1) Prove that for every natural numbers $m$, There is at most two primes $p$ where $p^{3}+m^{2}$ is the square of a number.
2) Find all ...
10
votes
2answers
197 views
$x^2+x+1$ is the cube of a prime.
Please help me find all natural numbers $x$ so that $x^2+x+1$ is the cube of a prime number.(Used in here)
3
votes
4answers
41 views
Proving $\frac{1}{2}(5x+4),\;2 < x,,\;\text{isPrime}(n)\Rightarrow n = 10k+7$
How is it possible to establish proof for the following statement?
$$n = \frac{1}{2}(5x+4),\;2<x,\;\text{isPrime}(n)\;\Rightarrow\;n=10k+7$$
Where $n,x,k$ are $\text{integers}$.
To be more ...
5
votes
3answers
208 views
What would be the immediate implications of a formula for prime numbers?
What would be the immediate implications for Math (or sciences as a general) if someone developed a formula capable of generating every prime number progressively and perfectly, also able to prove (or ...
2
votes
0answers
117 views
primes of the form $p=8k+1, 8k+3$ can be expressed as $p=a^2+2b^2$
I have trouble showing that primes of the form $p=8k+1, 8k+3$ can be expressed as $p=a^2+2b^2$.
Thanks in advance.
1
vote
0answers
28 views
Could a determinstic primality test specialized to this form of prime exist?
Is it possible there could be an "efficient" deterministic primality test for prime numbers of the form
$$(2^n + 1)^2 - 2$$
or
$$(2^n - 1)^2 - 2$$
in the same vein as the Lucas-Lehmer test for ...
3
votes
1answer
64 views
On the primality of integers of the form $p^2+k$
I am not able to find an answer to the following question:
For which positive even integers $k$ is the integer
$$p^2+k$$ prime, where $p$ is a prime number $\gt5$?
1
vote
0answers
33 views
Unique decomposition of $c$ sums of products of $k$ numbers greater than 1, allowing duplicates?
This question differs from Unique decomposition of $c$ sums of products of $k$ prime numbers, allowing duplicates? in that prime number restriction is changed to any number greater than 1.
Suppose ...
3
votes
0answers
67 views
Prime norm ideals that are also principal
Landau's prime number theorem tells us asymptotic formula for counting the number of prime ideals of a number field K, with norm at most X.
I am interested in the the prime ideals with a prime norm. ...
3
votes
1answer
50 views
$\sigma(\sigma(p^2)) \neq 2p^2$ for all odd primes $p$.
How to prove that $\sigma(\sigma(p^2)) \neq 2p^2$ for all odd primes $p$?
I know that $\sigma(p^2)=1+p+p^2$ but I can't progress anymore.
2
votes
3answers
40 views
Marking the prime points on a circle
If you travel around a circle and mark all the points on the circle where the distance you travelled is a prime number, where you would go through many rotations*, do you end up marking the entire ...
3
votes
2answers
47 views
Existence of a prime
If $x$ is odd and natural and ${x^2}+2\equiv3\mod 4$, how can I show there exists a prime $p$ such that $p|x^2+2$ and $p\equiv3\mod 4$.
1
vote
0answers
38 views
Unique decomposition of $c$ sums of products of $k$ prime numbers, allowing duplicates?
Suppose that there are $n$ different prime numbers. Define procedure a) as following ($k \leq n$ and $k$ fixed): procedure a) for each time, we select one number out of $n$ possible cases and multiply ...
1
vote
1answer
548 views
Conjecture on cycle length and primes
That thanks for Peter Košinár's answer,I change the conjecture a lot.
Suppose $a$ is positive odd, and $b = A179382((a+1)/2)$, if $b = (a-1)/(2^c)$ for some $c > 0$ with $a,b,c$ not multiples of ...
2
votes
1answer
49 views
The set of exponential primes
Consider a set of integers $Q$ such that the set of all positive integers $\mathbb{Z}$ is equivalent to the span of ever possible power tower
$$a_1^{a_2^{\ldots a_N}}$$ involving $a_i \in Q$.
In ...
2
votes
0answers
46 views
Making fermat's little theorem for composite numbers the ultimate test.
It is a programming question but mathematics has a major role to play in it.
I have to find the largest prime less than a number $n$. Note that $n\leq10^{18}$. I can go for Fermat's Little Theorem ...
0
votes
2answers
60 views
Is it true that $a^{k(p-1)+b} \;\stackrel{p}{\equiv} \;\;a^b\;$?
$$a^{k(p-1)+b} \;\stackrel{p}{\equiv} \;\;a^b\;?$$
$p$ prime number and $a,b,k\in\mathbb{N}^+$. And $p$ does not divide $a$.
According to Fermat's Little theorem $a^{p-1}\stackrel{p}{\equiv}1$. So ...
1
vote
0answers
49 views
Randomness in prime numbers
I'm very interested in possible randomness in prime numbers distribution. There are many methods for
"decomposition" regularity and randomness in primes (e.g. subtraction of some asymptotics , ...
10
votes
1answer
196 views
Can the order of 2 mod p be arbitrarily small (relative to $p - 1$)?
Given a prime number $p$, let $\operatorname{ord}_p(2)$ be the multiplicative order of $2$ modulo $p$, i.e., the smallest integer $k$ such that $p$ divides $2^k - 1$. By Lagrange's theorem, ...
1
vote
0answers
42 views
Consequencesof the Hadamard product expression of $L(s, \chi)$
I'm going through my lecture notes and I'm stuck on the proof of
For any $t>0$ and primitive $\chi$ modulo $q$
$$\sum_{\rho=\beta+i \gamma: \Lambda(\rho, ...
1
vote
0answers
63 views
Fourier Analysis of Prime Counting Function
I was thinking about the following:
Denote $\pi(x)$ as the prime counting function such that:
$$
\pi(x) = \#\text{ of prime numbers}\leq x
$$
It is well known from the prime number theorem that
$$
...
2
votes
2answers
74 views
What software can calculate the order of $b \mod p$, where $p$ is a large prime?
I wasn't sure where to ask this, but Mathematics seems better than StackOverflow or Programmers.
I have no background whatsoever in number theory, and I need to find software that can calculate the ...
12
votes
1answer
92 views
Is any closed-form representation known for the sum $\sum\limits_{n=1}^{\infty}\frac{\mu(n)\log n}{n^2}$?
Is any closed-form representation known for the sum $\sum\limits_{n=1}^{\infty}\frac{\mu(n)\log n}{n^2}$, where $\mu(n)$ is the Möbius $\mu$-function?
1
vote
1answer
39 views
A question about the Andrica's conjecture on the prime numbers
The Andrica's conjecture on the prime numbers states:
given a couple of prime numbers $p_k$ and $p_{k+1}$ the following inequality holds:
$$\sqrt{p_{k+1}}-\sqrt{p_{k}}\lt 1$$
Is it possible to show ...
1
vote
1answer
55 views
Does $a \mid bc$ imply $\frac{a}{(a,b)} \mid c$?
If $a \mid bc$, then does $\frac{a}{(a,b)} \mid c$? I doubt anybody here is industrious enough to show this via a diagram, but who knows.
5
votes
1answer
184 views
Is this number theory conjecture known to be true?
I've been working on proving that there is always a prime between $n$ and $2n$, and also that there is always a prime between $n^2$ and $(n+1)^2$ (Legendre's conjecture).
I believe I've proven those ...
5
votes
3answers
90 views
Size of largest prime factor
It is well known and easy to prove that the smallest prime factor of an integer $n$ is at most equal to $\sqrt n$. What can be said about the largest prime factor of $n$, denoted by $P_1(n)$? In ...
6
votes
2answers
63 views
Need to state “$p$ not equal to $61$” when solving $61p + 1 = n^2$?
In the pictures below, am I wrong to say that the 3 lines in the red box are not needed in the solutions? Regardless of whether 61 and p are distinct, it's still true that we have only the 2 possible ...
0
votes
1answer
47 views
Find if a number $n$ is a primitive root of $p$
Let $n = p_1\cdot p_2\cdot\ldots\cdot p_k$ where the $p_i$ are primes. Let $s = \varphi(n)$ where $\varphi$ denotes the Euler Totient Function.
If none of $p_1,p_2,\ldots,p_k$ makes $a^{(s/p_i)} = 1$ ...
11
votes
2answers
375 views
Has anyone found a “pattern” in prime numbers?
Yesterday I was having some fun trying to look for some patterns in primes; and I think I found something interesting (to me at least). I still have not found any lists of patterns already found, ...
5
votes
3answers
153 views
Integer solutions of $n^3 = p^2 - p - 1$
Find all integer solutions of the equation, $n^3 = p^2 - p - 1$, where p is prime.
5
votes
1answer
98 views
Understanding a famous proof by Jitsuro Nagura: Need help understanding one step in the main theorem
I am going through the proof by Jitsuro Nagura which shows that there is always a prime between $x$ and $\frac{6x}{5}$ where $x \ge 25$.
Nagura uses the following definitions:
$$\vartheta(x) = ...
3
votes
2answers
112 views
Prime numbers, what explains this pattern?
This morning I got a message on the Active Mathematica yahoo mailing list from the signature "in zero" asking to calculate this sum:
$$\sum _{k=1}^n \frac{\log (p_k)}{\log (p_n)}$$
where $p_n$ is ...
3
votes
2answers
84 views
Factorial primes
Factorial primes are primes of the form $n!\pm1$. (In this application I'm interested specifically in $n!+1$ but any answer is likely to apply to both forms.) It seems hard to prove that there are ...
2
votes
2answers
49 views
About linear combinations of primes
$a,b,c$ are natural numbers whose greatest common divisor is $1$.
$a,b,c\in\mathbb{N}^*$, $(a,b,c)=1$
Try to write down the expression using $a,b,c$ of the biggest natural number $M$ that cannot be ...
1
vote
0answers
26 views
conversion from psi function to prime counting function
Can we convert $\psi(x)$ to $\pi(x)$ without using integrals. Also if $\psi(x)>\psi(y)$
when we can say that $\pi(x)>\pi(y)$ . It seems that $\theta(x)>\theta(y)$ so $\pi(x)>\pi(y)$
but ...
2
votes
1answer
47 views
Inequality about prime numbers
I have been thinking about it lately. Let's think of prime number sequence: $$q_1,q_2,...q_n$$ where $q_1=2, q_2=3$ and onwards. Can we find an n such as the inequality $$q_n \gt ...
2
votes
0answers
29 views
$\sum_{i=1}^{k}[\sqrt{ip}]=\frac{p^2-1}{12}$, $p$ is a prime of the form $4k+1$ [duplicate]
$\sum_{i=1}^{k}[\sqrt{ip}]=\frac{p^2-1}{12}$, $p$ is a prime of the form $4k+1$
How to prove this?
4
votes
4answers
148 views
Why 4 is not a primitive root modulo p for any prime p?
I wonder why 4 is not a primitive root for any prime p ?
I've been trying to find an answer with no success so far. Any suggestion would be very helpful,
thanks in advance !
7
votes
2answers
131 views
Find all values x, y and z which satisfy the equation $(x^2 + 1)(y^2 + 1) = z^2 + 1$ given that $(x^2 + 1)$ and $(y^2 + 1)$ are both primes.
Find all positive integers x, y, z which satisfy the equation $(x^2 + 1)(y^2 + 1) = z^2 + 1$ given that $(x^2 + 1)$ and $(y^2 + 1)$ are both primes.
It seems trivial that the only set of integers x, ...
2
votes
3answers
76 views
Wiki proof of Lucas primality test
I have a question about one step in the proof:
Why does $a^{n-1} \equiv 1\ (\operatorname{mod} n)$ imply that $a$ and $n$ are coprime?
Thank you!
20
votes
0answers
392 views
Generalizing Ramanujan's proof of Bertrand's Postulate: Can Ramanujan's approach be used to show a prime between $4x$ and $5x$ for $x \ge 3$
Perhaps, I've been thinking too long about Ramanujan's proof, but it appears to me that his argument can be generalized beyond $x$ and $2x$. My argument below attempts to show that for $x \ge 1331$, ...
4
votes
2answers
51 views
Sum involving prime numbers
Given the series:
$$S=\sum_{k=1}^{N}\frac{k}{p_k}$$
where $p_k$ is the $k^{th}$ prime number, is it possible to know if this series converges in the limit:
$$\lim_{N\to\infty}S$$
and eventually, its ...
10
votes
2answers
135 views
About prime factor and consecutive integers
The problem is:
There exists an integer $N$ such that for any $n>N$, there exists $m \in \{n,n+1, \ldots ,n+9\}$ such that $m$ has at least $3$ distinct prime factors.
2 Years ago, My ...
3
votes
1answer
122 views
The meaning of the Euler Formula for Zeta
Does anybody know about a "meaning" behind the Euler Formula, what does it really say about the primes?
I know that it is in equation to the zeta function and also how it is derived, but cannot find ...
3
votes
1answer
67 views
Proof showing there exists a sequence of $m$ consecutive natural numbers which contains exactly $n$ primes.
Given that $n\in\Bbb N$, show that there exists a $k\in\Bbb N$ such that for all $m\ge k$, there exists a sequence of $m$ consecutive natural numbers which contains exactly $n$ primes.
1
vote
1answer
59 views
Dirichlets theorem on primes
I want to use Dirichlets theorem on primes for my diploma thesis. I want to use following form
Let $a,b\in\mathbb{N}$, such that $\gcd(a,b)=1$. Then the set $\{a\cdot n+b| n\in\mathbb{N}\}$ contains ...
