All Questions
24 questions
27
votes
1
answer
2k
views
The quaternion moat problem
"One cannot walk to infinity on the real line if one uses steps of bounded
length and steps on the prime numbers. This is simply
a restatement of the classic result that there are arbitrarily
...
- 151k
25
votes
7
answers
3k
views
Question on consecutive integers with similar prime factorizations
Suppose that $n=\prod_{i=1}^{k} p_i^{e_i}$ and $m=\prod_{i=1}^{l} q_i^{f_i}$ are prime factorizations of two positive integers $n$ and $m$, with the primes permuted so that $e_1 \le e_2 \cdots \le e_k$...
- 15.4k
22
votes
1
answer
2k
views
Are all primes in a PAP-3?
Van der Corput [1] proved that there are infinitely many arithmetic progressions of primes of length 3 (PAP-3). (Green & Tao [2] famously extended this theorem to length $k$.)
But taking this in ...
- 9,114
19
votes
2
answers
1k
views
Floors of rationals to powers: Infinite number of primes?
Let $r=a/b$ be a rational number in lowest terms, larger than $1$,
and not an integer (so $b > 1$).
Q. Does the sequence
$$ \lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor,
\...
- 151k
14
votes
1
answer
1k
views
Normal numbers, Liouville function, and the Riemann Hypothesis
This is a question about whether or not some number $\lambda^*$ is normal in base 2. More specifically, I am wondering if $\lambda^*$ is not normal. Proving it is normal would be next to impossible, ...
- 3,098
14
votes
2
answers
1k
views
Prime divisors of numbers 2^n + 3
I'm interested in the following problem: do there exist infinitely many prime numbers $p$ such that $p^2|2^{n}+3$ for some natural number $n$?
Some motivation:
If we replace the function $2^n + 3$ ...
- 191
13
votes
2
answers
3k
views
Does listing the prime factors always stop?
Take a natural number's prime factors and list them increasingly and repeating them according to multiplicity. Concatenate their decimal (or in any base) representation to get a new number and repeat ...
- 807
12
votes
1
answer
626
views
A conjecture by Euler about $8n+3$
Euler's conjecture: For any positive integer $n$, $8n+3$ can be represented as a sum
$$8n+3=(2k-1)^2+2p,$$
where $k$ is a positive integer, and $p$ is a prime.
I want to know whether there has been ...
- 121
12
votes
2
answers
2k
views
Detecting almost-primes quickly
There are many fast algorithms (deterministic and probabilistic) for detecting primality. Are there any fast algorithms (probabilistic ones allowed) known for detecting whether a number is the product ...
- 20.2k
11
votes
2
answers
2k
views
Distinctive property of the primes 17 and 19?
Consider the question whether it is true that a prime number $p$ divides
$1^1+2^2+3^3+....+(p-1)^{p-1}$ if and only if $p \in \{17,19\}$.
For the obvious heuristic reasons, for large $n$ one would ...
- 311
10
votes
1
answer
2k
views
What keeps asymptotic Goldbach's conjecture out of reach of current technology?
Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...
- 6,984
10
votes
1
answer
1k
views
Linear equation with primes
Is there an integer $n$ with an infinite number of representations of the form
$n=2q-p$, where $p$ and $q$ are both primes?
Given a positive integer $k>1$, I would like to know for which (if any) ...
- 566
6
votes
2
answers
804
views
Must Mersenne numbers be divisible by arbitrary large primes with exponent one?
Let $M_n$ denote the Mersenne numbers $M_n=2^n-1$.
As $n$ varies, must $M_n$ be divisible by arbitrary large prime $p$
with exponent one, i.e. $p \mid M_n, p^2 \nmid M_n$?
In other words, must the ...
- 25.4k
5
votes
1
answer
622
views
Ruth-Aaron triples, etc
A Ruth-Aaron pair is two numbers $(n,n+1)$ such that
their sum of prime factors is equal, counting repeated prime factors.
(The name refers to Hank Aaron's 715 homeruns surpassing Babe Ruth's 714!)
So
...
- 151k
5
votes
0
answers
667
views
Are there an infinite number of prime Euclid numbers?
A number defined as the product of first $n$ prime numbers $+1$ is called $n$th Euclid number. Are there any survey on the progress for answering the following question: are there an infinite number ...
3
votes
2
answers
2k
views
What is the importance of Polignac’s conjecture?
The twin-prime conjecture (also known as Polignac’s conjecture, 1846) states that there are infinitely many twin primes (pairs of primes that differ by 2; for example, 3 and 5, 5 and 7, 11 and 13, and ...
- 883
3
votes
1
answer
401
views
Probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$
Using my computer, I found that in the interval $[1, N]$ the probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$ is greater than constant $c$ where $N=10^2, 10^3,...,10^{9}$, $x$ ...
- 1,776
2
votes
4
answers
1k
views
Product of exponents of prime factorization
Let $p(n)$ be the product of the exponents of the prime factorization of $n$. For example,
$$p(5184) = p(2^6 3^4) = 24 \;,$$
$$p(65536) = p(2^{16}) = 16 \;.$$
Define $P(n)$ as the number of iterations ...
- 151k
2
votes
0
answers
157
views
Conjecture: $x^4+1$ is never Wieferich prime
Related to this question and Alexander Kalmynin's answer.
For natural $n$ define $J(n)=(2^{n-1}-1) \bmod n^2$
and if $n$ is power of two define $J(2^n)=1$ (this is artificial, just to
avoid triviality ...
- 25.4k
1
vote
1
answer
325
views
Goldbach conjecture reformulation [closed]
As thought, the question below is a reformulation of the goldbach conjecture.
$ S = \{K - ap \mid a \geq 3, p \text{ is prime} < K/2 \} $, where $ a $ is an odd integer greater than or equal to 3, ...
- 23
1
vote
1
answer
190
views
Infinitely many $k \in \mathbb{N}$ such that the closed interval $[k, k+99]$ contains from $2$ to $23$ prime numbers
Let $k \in \mathbb{Z}^+$.
Is it possible to prove that, for some given
$m \in \{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23\}$,
there are only finitely many $k$ such that the closed ...
- 1,451
1
vote
0
answers
243
views
Is there a known connection between Wieferich primes and the Goormaghtigh conjecture?
I posted this question on SE, and was told I should repost it here.
The Goormaghtigh conjecture explores the Diophantine equation of the form
$$
\frac{a^b-1}{a-1}=\frac{c^d-1}{c-1},
$$
where $a>c&...
- 119
0
votes
1
answer
124
views
Could I possibly exploit distinct odd primes raised to 6 to solve Exact Three Cover, when reducing it in Subset Sum?
I'm solving Exact 3 Cover, given a list with no duplicates $S$ of $3m$ whole numbers and a collection $C$ of subsets of $S$, each containing exactly three elements. The goal is to decide if there are $...
- 101
0
votes
0
answers
110
views
What will be the set of non-Wieferich numbers if the set of non-Wieferich primes is finite?
Integer $n$ is Wieferich number if $2^{\phi(n)}-1 \equiv 0 \pmod {n^2}$.
Wieferich prime is Wieferich number with $n$ prime.
It is an open problem if there are infinitely many Wieferich primes
and ...
- 25.4k