**0**

votes

**1**answer

380 views

### A possible consequence of Dirichlet's theorem about primes in arithmetic progression

EDIT : I copy-paste the beginning of a previous question since Gerry Myerson suggested this question should be self-contained.
"let's consider a composite natural number $n$ greater or equal to $4$. ...

**8**

votes

**0**answers

529 views

### Would the following conjectures imply $\lim\inf_{n\to\infty}p_{n+k}-p_{n}=O(k\log k)$?

Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, ...

**61**

votes

**6**answers

6k 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 ...

**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).
...

**13**

votes

**4**answers

1k views

### Arithmetic progressions without small primes

The following question came up in the discussion at How small can a group with an n-dimensional irreducible complex representation be? :
Is it known that there are infinitely many primes p for which ...

**4**

votes

**3**answers

504 views

### Bound the error in estimating a relative totient function

Let $n=p_1^{e_1}\cdots p_k^{e_k}$ be an integer with $k$ prime factors. We know that the number of integers less than $n$ and coprime to it is
$$\Phi(n)=n-\sum_i\frac n{p_i}\+\sum_{i \lt j}\frac ...

**39**

votes

**4**answers

2k 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 ...

**31**

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 ...

**17**

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 ...

**16**

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 ...

**15**

votes

**1**answer

655 views

### Sums of primes that are themselves prime

I'm not a math expert so this may be a trivial question; if $p_i$ is the $i$-th prime, let:
$$S(n) = \sum_{i=1}^n p_i$$
be the sum of the first $n$ primes and
$$P(n) = | \{1 \leq i \leq n \mid ...

**12**

votes

**1**answer

538 views

### Analytic lower bounds on the first sign change of pi(x) - li(x)?

There have been many results on the first sign change of $\pi(x)-{\mathrm{li}}(x)$: among others, Lehman, te Riele, Bays & Hudson, Demichael, Chao & Plymen, and most recently Saouter & ...

**6**

votes

**6**answers

749 views

### Sequences equidistributed modulo 1

Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results.
H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidistributed modulo 1.
...

**8**

votes

**6**answers

1k views

### Are there any interesting or lesser known proofs related to Bertrand's Postulate

There are 3 standard proofs of Bertrand's Postulate:
(1) Chebyshev's original proof
(2) Ramanujan's simplification of Chebyshev's proof
(3) Erdos's proof
I recently learned about the very ...

**7**

votes

**2**answers

413 views

### A non-standard ergodic limit

Suppose $T$ is an ergodic measure-preserving transformation on a measure space $(X,\Sigma,\mu)$, and $f\in L^1(\mu)$. Does the limit
$\lim_{X\to\infty} \pi(X)^{-1}\sum_{p\leq X} f(T^{p}x)$
exist ...

**10**

votes

**1**answer

2k views

### The Green-Tao theorem and positive binary quadratic forms

Some time ago I asked a question on consecutive numbers represented integrally by an integral positive binary quadratic form. It has occurred to me that, instead, the Green-Tao theorem may include a ...

**6**

votes

**2**answers

534 views

### At what point would an elementary generalization of Bertrand's Postulate be interesting?

I know that in 1952 Jitsuro Nagura was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k > 24$.
At what point would an improvement on Nagura's result be interesting? ...

**6**

votes

**3**answers

587 views

### a question for the prime counting function

A famous inequality that has been proved by J.B. Rosser and L. Schoenfeld says that
$\frac{n}{\ln n-1/2}$ < $\pi(n)$<$\frac{n}{\ln n-3/2} , n\ge 67$.
Using this inequality we can prove ...

**6**

votes

**3**answers

855 views

### Values where infinite products of primes and composites are equal

Highly grateful for your help/steers on the following question (at the end):
Take the infinite product:
$$\displaystyle T(s) = \prod _{n=2}^{\infty } \left( \dfrac{{n}^{s}} {{n}^{s}-1}\right)$$
for ...

**18**

votes

**0**answers

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 ...

**7**

votes

**1**answer

503 views

### Least prime primitive root

For $p$ a prime number, let $G(p)$ be the least prime $q$ such that $q$ is a primitive root mod $p$, that is $q$ generates the multiplicative group $(\mathbb Z/p\mathbb Z$)* .
Is it known that ...

**7**

votes

**3**answers

1k views

### Can the twin prime problem be solved with a single use of a halting oracle?

It occurred to me that if it were possible to determine whether a given program halts, that could be used to answer the twin primes conjecture
A) Write a program which takes input n and then counts ...

**4**

votes

**1**answer

280 views

### A “bit” of primes

Is there anything known/proved/conjectured about the distribution of:
$$B(n) = \frac{(p_n-1)}{2} \bmod 2, \qquad p_n \mbox{ is the } n\mbox{-th prime}$$
i.e. the bit 1 of the binary representation ...

**4**

votes

**5**answers

1k views

### residue classes of primes, covering intervals and bounds on the different ways

Take the first $n$ primes $p_1,...,p_n$ and the primorial $P_n$ .Denote by $p_i$ every prime bigger than $p_n$ and smaller than $P_n$.
1) Is that true that there always be a number in any interval of ...

**1**

vote

**1**answer

112 views

### Are there any theorems about a prime $p > k$ in a sequence stronger than Sylvester-Schur?

Sylvester-Schur says: "if $n \ge 2k$, then there is a number in the list
$n − k + 1, n − k + 2,$ ... $, n$
divisible by a prime $p > k$."
Shouldn't it also be true that if $n \ge k$, then there is ...

**9**

votes

**2**answers

496 views

### Odd-bit primes ratio

Say that a number is an odd-bit number if
the count of 1-bits in its binary representation is odd.
Define an even-bit number analogously.
Thus $541 = 1000011101_2$ is an odd-bit number,
and $523 = ...

**7**

votes

**1**answer

408 views

### The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$

The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
...

**6**

votes

**0**answers

565 views

### Would the following conjectures imply Cramer's conjecture?

Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, ...

**4**

votes

**1**answer

454 views

### p such that p+1 has a large prime factor, effectively

I was reading the Boneh-Franklin IBE paper, and it seemed rather conspicuous to me that they
didn't address the question of how to find primes $p$ and $q$ satisfying what they need (on page 19).
...

**3**

votes

**1**answer

456 views

### Work down on the Upper bound of the Twin Primes [duplicate]

It can be shown using the Selberg Sieve method, that the maximum number of Twin primes less than $N$ is
$$\frac{CN}{\ln^2(N)}$$
does anyone know if there has been any work done on finding an upper ...

**1**

vote

**3**answers

201 views

### Powers of $2$ and the products of initial odd primes

NOTATION: $O_x$ -- the product of all odd primes $\le x$.
E.g. $O_7=3\cdot 5\cdot 7 = 105$.
QUESTION: Are the three ordered pairs $\ (d\ p)=(1\ 3)\ \ (2\ 3)\ \ (4\ 5)\ $ the only solutions of the ...