# Tagged Questions

**16**

votes

**1**answer

544 views

### How many integers divide a number that involves just three non-zero digits?

Just to be concrete, consider the digits to be binary. Hasse showed that among all the primes, only a fraction of $17/24 < 1$ divide a number of the form $2^n+1$. As a result, the integers that ...

**7**

votes

**0**answers

164 views

### In which orders can the numbers of prime factors of consecutive integers be?

Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur.
Given ...

**3**

votes

**1**answer

235 views

### Ratio of consecutive divisors and average

Let $2\leq d_1 < d_2,...,d_l < n$ be all the proper nontrivial divisors of $n$. I like to understand how much these divisors deviates from each other. Here are two questions in this regard:
(1) ...

**4**

votes

**1**answer

264 views

### Distribution function for divisors of an Integer

For a fixed $n$, let $D_n(x) = \{ d|n : d \leq x \}$ . We assume here $p \leq x \leq n/p$,
where $p$ is the smallest prime factor of $n$.
For example if $n = p^i$ for some prime $p$ then $D_n(x) ...

**3**

votes

**2**answers

395 views

### using distribution of primes to generate random bits?

In his popular science book The Music of the Primes, Marcus du Sautoy tries to link the truth of the Riemann Hypothesis to the "randomness" of the primes. To do this, he invokes the idea of a "fair ...

**2**

votes

**1**answer

198 views

### Has every divisibility-antichain density zero?

Let $A \subset \mathbb N$ be a antichain with respect to divisibility. Does this imply that the density of $A$ is $0$?

**21**

votes

**2**answers

905 views

### Drawing natural numbers without replacement.

Suppose we start with an initial probability distribution on $\mathbb{N}$ that gives positive probability to each $n$. Let's call this random variable $X_1$ so we have $P(X_1=n)=p_{1,n}>0$ for all ...

**3**

votes

**1**answer

310 views

### Work exploring application of probability to metric number theory problems

I am interested in studying the application of probabilistic tools to study metric number theoretic problems, specifically the Duffin-Schaeffer conjecture ...

**21**

votes

**2**answers

1k views

### Is there any sense in which Dirichlet density is “optimal?”

A philosopher asked me an interesting math question today! We know that there are sets S of integers which don't have a "natural" or "naive" density -- that is, the quantity (1/n)|S intersect [1..n]| ...

**11**

votes

**1**answer

1k views

### Heuristic reason for Polya's conjecture

Let $\lambda(n)$ be Liouville's function, so that for each positive integer $n = p_1^{m_1}\cdots p_r^{m_r}$, we have that $\lambda(n) = (-1)^{\sum^{r}_{k=1}{m_k}}$. In 1919, Polya conjectured that ...

**15**

votes

**1**answer

521 views

### What's the probability that k + n^2 is squarefree, for fixed k?

While playing around with this question (when is the sum of two squares squarefree?), from some experimental computations (and bolstered by the fact that the density of squarefree positive integers is ...