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Results tagged with prime-numbers
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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
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1
answer
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Does the mean ratio of the largest prime factor in prime gaps to the lower bound of the gap ...
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
20
votes
2
answers
2k
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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 numb …
- 114k
3
votes
0
answers
163
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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 whos …
- 5,470
15
votes
0
answers
361
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Do primes of the form $4k+1$ ever lead the greatest prime factor race?
Analogous to Chebyshev's race between primes, I examined the race between primes in the greatest prime factors, GPF, of natural numbers. Similar to the regular prime race, in the GPF race, the proport …
- 12.9k
6
votes
4
answers
2k
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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 …
2
votes
0
answers
297
views
How soon can we represent a number as the sum of two primes?
Posting in MO since it was unanswered in MSE.
Goldbach's conjecture says that every even number can be represented as the sum of two primes. But how soon can we find such a representation. Taking $20 …
4
votes
1
answer
217
views
Numbers $n$ whose representation as the product of two divisors require more digits than tha...
Note: Posting in MO since it was unanswered in MSE
Let $f(x)$ be the number of digits in the decimal representation of $x$ e.g. $, f(0) = 1, f(1729) = 4$. If $n = ab$ then we can show that $f(ab) > f( …
- 645
9
votes
3
answers
581
views
Why is there an unexpected increase in the density of certain types of Goldbach primes?
Note: Posted in MO since it was unanswered in MSE.
I was checking how quickly we can verify Goldbach's conjecture for a given even number $n$ and it was clear that searching backward starting from the …
- 30.1k
4
votes
1
answer
234
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Relation between $\pi$, area and the sides of Pythagorean triangles whose hypotenuse is a pr...
Consider all Pythagorean triangles $a^2 + b^2 = p^2$ in which the hypotenuse $p$ is a prime number. Let $h(x) = \sum_{p \le x}p^2$, $a(x) = \sum_{p \le x}ab$ and $r(x) = \sum_{p \le x}(a+b)^2$. Is it …
- 188k
2
votes
0
answers
132
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Average length of consecutive integers which have an increasing number of divisors
Consider the nine consecutive natural numbers starting from $1584614377$.
n = 1584614377 no. of divisors: 2
n = 1584614378 no. of divisors: 4
n = 1584614379 no. of divisors: 8
n = 1584614380 no. of di …
- 5,470
8
votes
1
answer
826
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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 …
- 114k
2
votes
1
answer
306
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Analogue of Fermat's little theorem for Bernoulli numbers
Is the following analogue of Fermat's Little Theorem for Bernoulli numbers true?
Let $D_{2n}$ be the denominator of $\frac{B_{2n}}{4n}$ where $B_n$ is
the $n$-th Bernoulli number. If $\gcd(a, D_{2n}) …
- 63.9k
7
votes
1
answer
370
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If $n = 18k+5$ is composite, there are at least 9 divisors of $\phi(n)$ which do not divide ...
If $n$ is a composite of the form $18k+5$, there at least 9 divisors of $\phi(n)$ which do not divide $n-1$. Is this true in general or if not, what is the smallest counter example? The conjecture has …
CommunityBot
- 1
5
votes
1
answer
958
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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 …
- 3,432
1
vote
0
answers
133
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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 …
- 5,470