4
votes
1answer
422 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 ...
17
votes
2answers
702 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, ...
3
votes
4answers
625 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 ...
9
votes
1answer
492 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 ...
12
votes
2answers
1k 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 ...
4
votes
1answer
413 views

Implication of Polignac's conjecture on prime distribution in models of PA

Polignac's conjecture (PC) is that there exists infinitely many pairs of consecutive prime numbers that are a distance $d$ apart for some natural number $d$. The twin prime conjecture is the ...
16
votes
1answer
751 views

Covering the primes by arithmetic progressions

Define the length of a set of arithmetic progressions of natural numbers $A=\lbrace A_1, A_2, \ldots \rbrace$ to be $\min_i | A_i |$: the length of the shortest sequence among all the progressions. ...
4
votes
0answers
499 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 ...
15
votes
2answers
2k views

Walking to infinity on the primes: The prime-spiral moat problem

It is an unsolved problem to decide if it is possible to "walk to infinity" from the origin with bounded-length steps, each touching a Gaussian prime as a stepping stone. The paper by Ellen Gethner, ...
13
votes
2answers
965 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 ...
16
votes
1answer
1k 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 ...
16
votes
6answers
2k 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 ...
11
votes
2answers
2k 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 ...
12
votes
2answers
786 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 $2^n + 3$ ...
9
votes
1answer
774 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) ...