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Results tagged with prime-numbers
Search options questions only
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user 23388
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
11
votes
1
answer
926
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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 …
- 5,470
2
votes
0
answers
131
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 …
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4
votes
2
answers
699
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 …
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6
votes
3
answers
759
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} + …
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3
votes
3
answers
803
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 …
- 5,470
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 …
- 5,470
5
votes
0
answers
89
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 …
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3
votes
0
answers
322
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 …
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1
vote
0
answers
133
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 …
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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)}.
$$
I am looking at a special case of thi …
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8
votes
1
answer
826
views
Are there highly composite prime gaps?
Definition: Highly composite prime gap
The three composite numbers between the consecutive primes $643$ and $647$ each have at least three distinct prime factors. This is the first occurrence of prime …
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5
votes
1
answer
958
views
There at least 4 divisors of $n-1$ which do not divide $\phi(n)$ if $n$ is a composite of th...
If $n$ is composite then $\phi(n) < n-1$ (Euler's totient function) hence there must be one or more divisors of $n-1$ which do not divide $\phi(n)$. For lack of a better terminology, let us call these …
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7
votes
0
answers
274
views
Are there infinitely many zeroes of $\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) $?
Let $\mu(n)$ be the Möbius function and $S(x)$ be the number of positive integers $n \le x$ such that
$$
\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) = 0
$$
My experimental data for $n \le 6 \times 10^5 $se …
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11
votes
1
answer
434
views
How many numbers $\le x$ can be factorized into three numbers which form the sides of a tria...
Note: Posting in MO since it was unanswered in MSE
Definition: We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3, 1 \le d_1 \le d_2 \l …
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6
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 …
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