**90**

votes

**5**answers

5k views

### Gaussian prime spirals

Imagine a particle in the complex plane, starting at $c_0$, a Gaussian integer,
moving initially $\pm$ in the horizontal
or vertical directions. When it hits a Gaussian prime, it turns left ...

**80**

votes

**4**answers

28k views

### Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture

Yitang Zhang recently published a new attack on the Twin Primes Conjecture. Quoting Andre Granville :
“The big experts in the field had
already tried to make this approach
work,” Granville ...

**65**

votes

**6**answers

7k views

### Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?

Let $p_n$ be the $n$-th prime number, as usual:
$p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$, etc.
For $k=1,2,3,\ldots$, define
$$
g_k = \liminf_{n \rightarrow \infty} (p_{n+k} - p_n).
$$
Thus the twin ...

**52**

votes

**0**answers

3k views

### The topology of Arithmetic Progressions of primes

The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner ...

**48**

votes

**4**answers

4k views

### Strange (or stupid) arithmetic derivation

Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...

**43**

votes

**3**answers

6k views

### Is a “non-analytic” proof of Dirichlet's theorem on primes known or possible?

It is well-known that one can prove certain special cases of Dirichlet's theorem by exhibiting an integer polynomial $p(x)$ with the properties that the prime divisors of $\{ p(n) | n \in \mathbb{Z} ...

**40**

votes

**1**answer

8k views

### Is the Green-Tao theorem true for primes within a given arithmetic progression?

Ben Green and Terrence Tao proved that there are arbitrary length arithmetic progressions among the primes.
Now, consider an arithmetic progression with starting term $a$ and common difference $d$. ...

**39**

votes

**1**answer

1k views

### Exploding primes

Suppose every prime $n$ could "explode" once.
An explosion results in $\lfloor \alpha \ln n \rfloor$ particles being
uniformly distributed over the integers in a range $n \pm \lfloor \beta \ln n ...

**37**

votes

**1**answer

3k views

### Are the primes normally distributed? Or is this the Riemann hypothesis?

Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes.
Let $\mathrm{Li}(x)$ be the offset logarithmic ...

**35**

votes

**4**answers

8k views

### How hard is it to compute the number of prime factors of a given integer?

I asked a related question on this mathoverflow thread. That question was promptly answered. This is a natural followup question to that one, which I decided to repost since that question is answered.
...

**34**

votes

**2**answers

2k views

### A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself

Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...

**34**

votes

**5**answers

2k views

### Happy New Prime Year!

It happens that next year 2011 is prime, while outgoing 2010 is
highly composite in the sense that the number of its distinct prime factors
is 4, maximal possible for a year $< 2310$.
Let me ...

**32**

votes

**3**answers

3k views

### Why could Mertens not prove the prime number theorem?

We know that
$$
\sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x)
$$
where $c_1$ is a constant. Again Mertens' theorem says that the primes $p$ satisfy
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln ...

**32**

votes

**4**answers

2k views

### Are There Primes of Every Hamming Weight?

That is, for every integer $n \in \mathbb{Z}_{>0}$ does there exist a prime which is the sum of $n$ distinct powers of $2$?
In this case, the Hamming weight of a number is the number of $1$s in ...

**31**

votes

**1**answer

3k views

### Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $

EDIT, 9 March 2014: when I asked this in 2010, I did not have the courage of my convictions, and so did not ask for an if and only if proof, as Kevin Buzzard quite properly pointed out. Such problems ...

**31**

votes

**2**answers

1k views

### Does iterating a certain function related to the sums of divisors eventually always result in a prime value?

Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$.
For example $f(6)=6+3+2=11$, $f(5)=5$.
Note that $x$ is a fixed point for ...

**29**

votes

**2**answers

2k views

### Generalization of Tamarkin's ARO 1993, final round, problem 10/8: still a conjecture?

This is from the category "problems I cannot believe that are still open". But then again, I don't know whether it is still open; it seems to have escaped the attention of most number theorists and ...

**29**

votes

**1**answer

943 views

### Prime Number Races in 2 Dimensions

Is the mapping $$f: \ \mathbb{N} \rightarrow \mathbb{Z}[i], \ \ \ n \ \mapsto
\sum_{2 < p \leq n \ {\rm prime}} e^{\frac{p-1}{4} \pi i}$$ surjective?
In 1999, when I was an undergraduate student, ...

**28**

votes

**2**answers

3k views

### Is Li(x) the best possible approximation to the prime-counting function?

The Prime Number Theorem says that $\lim_{n \to \infty} \frac{\pi(n)}{\mathrm{Li}(n)} = 1$, where $\mathrm{Li}(x)$ is the Logarithm integral function $\mathrm{Li}(x) = \int_2^x \frac{1}{\log(x)}dx$. ...

**28**

votes

**3**answers

2k views

### What is the current status of Agrawal's conjecture?

In their famous 'Primes is in P' paper Agrawal, Kayal and Saxena stated the following conjecture:
If for coprime integers $n$ and $r$ the equality $(X-1)^n = X^n - 1$ holds in ...

**27**

votes

**4**answers

3k views

### What is exceptional about the prime numbers 2 and 3?

Admittedly this question is vague. But I hope to convey my point. Feel free to downvote this.
Permit me to define prime number the following way:
A number $n>1$ is a prime if all integers $d$ ...

**26**

votes

**3**answers

3k views

### Bringing Number and Graph Theory Together: A Conjecture on Prime Numbers

Some MOers have been skeptic whether something like natural number graphs can be defined coherently such that every finite graph is isomorphic to such a graph. (See my previous questions [1], [2], ...

**24**

votes

**3**answers

2k views

### Composite pairs of the form n!-1 and n!+1

It's well known that the numbers of the form $n!\pm1$ are not always prime. Indeed, Wilson's Theorem guarantees that $(p-2)!-1$ and $(p-1)!+1$ are composite for every prime number $p > 5$.
Is ...

**23**

votes

**3**answers

1k views

### What does the numerically verified part of the Riemann Hypothesis tell about prime numbers?

I'm curious about the following question:
As of 2005(?) the Riemann hypothesis is verified for the first 10 trillion zeroes, they are all on the critical line. Does this verification gives us any ...

**22**

votes

**2**answers

3k views

### Why do primes dislike dividing the sum of all the preceding primes?

I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I ...

**22**

votes

**1**answer

1k views

### Permutations of $(Z/pZ)^*$

Let $p$ be a prime integer, and let $(\mathbb Z/p\mathbb Z)^*$ be the set of non-zero elements of $\mathbb Z/p \mathbb Z$.
Denote by $S((\mathbb Z/p \mathbb Z)^*)$ the group of permutations of ...

**21**

votes

**8**answers

10k views

### What are the connections between pi and prime numbers?

I watched a video that said the probability for Gaussian integers to be relatively prime is an expression in $\pi$, and I also know about $\zeta(2) = \pi^2/6$ but I am wondering what are more ...

**21**

votes

**4**answers

2k views

### Why so difficult to prove infinitely many restricted primes?

I wondered whether there were an infinite number of
palindromic primes written in binary (11, 101, 111, 10001, 11111, 1001001, 1101011, ...)
and quickly discovered that it is unknown
(OEIS A117697).
...

**21**

votes

**2**answers

1k views

### What is the crucial difference the Maynard/Tao approach and Goldston-Pintz-Yildirim that extends to prime k-tuples with $k>2$

Suppose $m$ is a positive integer. A quantity of interest is
$$
H_m = \liminf_{n\to\infty} \left(p_{n+m} - p_n \right)
$$
The twin prime conjecture, is, of course $H_1 = 2$, the the prime k-tuples ...

**21**

votes

**3**answers

775 views

### Are sets with similar asymptotic behavior as the primes necessarily finite additive bases?

The set of primes $\mathbb{P}$ has many interesting properties in additive number theory and some of the most famous open problems about $\mathbb{P}$ are the well-known Goldbach's strong and weak ...

**21**

votes

**3**answers

2k views

### Constructing prime numbers

The classical proof of the infiniteness of prime numbers is to take the $k$ first prime numbers $p_1,\ldots,p_k$, then to form
$$n_k:=1+p_1\cdots p_k.$$
Then $n_k$ has a prime factor, which is none of ...

**20**

votes

**6**answers

2k views

### Philosophical Question related to Largest Known Primes

The other day while discussing math, and primes specifically, the following question came to mind, and I figured I'd ask it here to see what people's opinions on it might be.
Main Question: ...

**20**

votes

**4**answers

4k views

### How does Yitang Zhang use Cauchy's inequality and Theorem 2 to obtain the error term coming from the $S_2$ sum

I have been reading Yitang Zhang's paper now for one and a half weeks and also volunteered to give a popular talk on the paper next week at Stockholm University.
Today I found a detail in the proof ...

**20**

votes

**6**answers

3k views

### explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros.
Because there are various explicit formulae ...

**20**

votes

**1**answer

1k views

### Nontrivial circular arguments?

There is a famous circular argument for the Prime Number Theorem (PNT). It turns
out that there exists an infinite sequence of elementary-to-prove Chebyshev-type estimates
that taken together imply ...

**19**

votes

**7**answers

2k views

### Asymptotic density of k-almost primes

Let $\pi_k(x)=|\{n\le x:n=p_1p_2\cdots p_k\}|$ be the counting function for the k-almost primes, generalizing $\pi(x)=\pi_1(x)$. A result of Landau is
$$\pi_k(x)\sim\frac{x(\log\log ...

**19**

votes

**1**answer

530 views

### Avoiding multiples of $p$

Let $p$ be a prime number and $P=\{1,2,...,p-1\}$
In how many ways we can sum all the elements of $P$ in such a way that we will reach a multiple of $p$
only when we sum the last summand?
For ...

**18**

votes

**2**answers

732 views

### Why are this operator's primes the Sophie Germain primes?

I was seeking a binary operator on natural
numbers that is intermediate between
the sum and the product, and explored this natural
candidate:
$$x \star y = \lceil (x y + x + y)/2 \rceil \;.$$
Then ...

**18**

votes

**2**answers

1k views

### The prime numbers modulo $k$, are not periodic

Consider the sequence of prime numbers: $2,3,5,7, \cdots$. Now reduce this sequence modulo $k$ for some integer $k > 2$. Show the resulting sequence is not periodic. :
EDIT: As noted in the ...

**18**

votes

**4**answers

1k views

### Generalizing Euclid's proof of the infinity of primes

Is the following problem still open? Let $S$ be a non-empty set of prime numbers such that whenever $p,q\in S$, all the prime factors of $pq+1$ are also elements of $S$. Is $S$ infinite?

**18**

votes

**3**answers

907 views

### Finite sums of prime numbers $\geq x$

Let $S_x$ be the set of finite sums of prime numbers $\geq x$. In other words, let $S_x$ be the submonoid of $(\mathbf{Z}_{\geq 0},+)$ generated by the set $\mathcal{P}_{\geq x}$ of prime numbers ...

**18**

votes

**3**answers

789 views

### Can Gauss sums derandomize any heuristic arguments?

I've always been fascinated by the fact that the classical Gauss sum has absolute value $\sqrt p$, which is exactly what we would expect if we were to interpret the Gauss sum as a random walk. In ...

**18**

votes

**1**answer

1k 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
large ...

**17**

votes

**6**answers

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 ...

**17**

votes

**2**answers

1k views

### Primes of the form $x^2+ny^2$ and congruences.

The answer of following classical problem is surely known, but I can't find a reference
For which positive integer $n$ is the set $S_n$ of primes of the form $x^2+n y^2$ ($x$, $y$ integers) ...

**17**

votes

**3**answers

1k views

### A binomial sum is divisible by p^2

This is a question I have since longer time, but I have absolutely no idea how to proceed on it.
Let $p>3$ be a prime. Prove that ...

**17**

votes

**1**answer

738 views

### The conjecture of Montgomery and Soundararajan on primes in short intervals: Empirical inconsistencies?

Assume that $y/ \log x \rightarrow \infty$ and that $y/x \rightarrow 0$. Then, from a conjecture by Montgomery and Soundararajan, we expect the number of primes in the interval $[x,x+y]$ to be ...

**17**

votes

**2**answers

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, ...

**17**

votes

**2**answers

743 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,
...

**17**

votes

**3**answers

1k views

### Primes of the form $x^2+ny^2+mz^2$ and congruences.

This is a sequel of this question where I asked for which positive integer $n$ the
set of primes of the former $x^2+ny^2$ was defined by congruences (a set of primes $P$ is defined by congruences if ...