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Results tagged with nt.number-theory
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user 23388
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
17
votes
2
answers
699
views
Does the mean ratio of the perimeter to the hypotenuse of right triangles converge to $1 + \...
Conjecture: Let $\mu_x$ be the arithmetic mean of the ratio of the
perimeter to the hypotenuse of all primitive Pythagorean triplets in
which no side exceeds $x$; then,
$$ \lim_{x \to \i …
- 4,628
6
votes
3
answers
709
views
A simple looking problem in partitions that became increasingly complex
I began with problem which looked simple in the beginning but became increasingly complex as I dug deeper.
Main questions: Find the number of solutions $s(n)$ of the equation
$$
n = \frac{k_1}{1} + …
- 4,628
11
votes
1
answer
785
views
Riemann sum formula for definite integral using prime numbers
I had asked this question in MSE. It got lot of upvotes but no answer (except one which was too long to be posted as a comment) hence I am posting it in MO.
While answering another question in MSE I …
- 4,628
4
votes
2
answers
689
views
On a sum involving prime numbers
I find myself needing the asymtotics of the following summation for my work. Let $a$ be a positive real number and $p_n$ be the $n$-th prime.
$$
\sum_{k=1}^{n} [k^a - (k-1)^a]p_k
$$
At $a=1$, this …
- 4,628
2
votes
0
answers
130
views
How many divisors of $\phi(m)$ do not divide $m-1$?
Lehmer's totient problem asks if there exists a composite number $m$ such that $\phi(m)$ divides $m-1$. Lower bounds on $m$ has been established but we do not know if a solution exists. Clearly, if we …
- 4,628
3
votes
3
answers
765
views
Mathematical techniques to reduce the amount of storage memory
Apologies for the length question. Those acquainted with the analytics industry will know that the next big thing in the information technology world will be the Big Data revolution where huge volumes …
- 4,628
8
votes
1
answer
746
views
Are there highly composite prime gaps?
Definition: Highly composite prime gap
The three composite numbers between the consecutive primes $643$ and $647$ each have at least three distinct prime factors. This is the first occurrence of prime …
- 4,628
5
votes
6
answers
2k
views
Sequences equidistributed modulo 1
Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results.
H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidistributed modulo 1.
I …
- 4,628
23
votes
1
answer
3k
views
Does the average primeness of natural numbers tend to zero?
This question was posted in MSE. It got many upvotes but no answer hence posting it in MO.
A number is either prime or composite, hence primality is a binary concept. Instead I wanted to put a valu …
- 4,628
1
vote
0
answers
46
views
Mean value of a function with binomial coefficients as weights
Is the following true?
Let $a$ be a positive integer and let $t_n$ be a sequence of numbers. We define the binomial mean of $t_n$
$$
\beta_{t_n,a} = \frac{1}{2^n t_n}\sum_{r^a \le n} \binom{n}{r^a …
- 4,628
5
votes
0
answers
87
views
Is the ratio of a number to the variance of its divisors injective?
The variance $v_n$ of a natural number $n$ is defined as the variance of its divisors. There are distinct integer whose variances are equal e,g. $v_{691} = v_{817}$. However I observed that for $n \le …
- 4,628
3
votes
0
answers
309
views
If $p^2 - q^2$ is a perfect square where $p$ and $q$ are primes $> 5000$ then is one of its ... [closed]
Is it true that if $p^2 - q^2$ is a perfect square where $p$ and $q$ are primes $> 5000$ then it has a prime factor greater than $17$?
Note: This question was asked in MSE but did not receive an answ …
- 4,628
6
votes
4
answers
2k
views
Probability that randomly chosen integers from a restricted set of natural numbers are coprime
We know that the probability $P(k)$ of $k$ randomly chosen integers $(k \ge 2)$ from the set of natural number are coprime is
$$
P(k) = \frac{1}{\zeta(k)}.
$$
I am looking at a special case of thi …
- 4,628
7
votes
0
answers
268
views
Are there infinitely many zeroes of $\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) $?
Let $\mu(n)$ be the Möbius function and $S(x)$ be the number of positive integers $n \le x$ such that
$$
\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) = 0
$$
My experimental data for $n \le 6 \times 10^5 $se …
- 4,628
1
vote
0
answers
119
views
Primes which do not divide certain homogeneous polynomials
It is known that if $x^2 + y^2 = z^2$ is a primitive Pythagorean triplet then $z$ is not divisible by any prime of the form $4k-1$. The following is a generalization of this classical result which sh …
- 4,628