**5**

votes

**3**answers

265 views

### Quadratic residues and nonresidues of arbitrary patterns

Let $p_1, p_2, \dotsc, p_n$ be distinct primes, and let $\epsilon_1, \epsilon_2, \dotsc, \epsilon_n$ be an arbitrary sequence of $1$ and $-1$.
There is an integer $a$ such that $\left( \frac{a}{p_1} ...

**2**

votes

**0**answers

212 views

### Efficient ways to count primes satisfying Zhang's theorem

The theorem of Yitang Zhang states that there exist a finite $k \in \mathbb{N}$ such that there exist infinitely pairs of primes $(p,q)$ such that $|p - q| \leq k$. The statement that $k$ can be taken ...

**6**

votes

**2**answers

493 views

### On the prime number theorem in arithmetic progression

The prime number theorem tells us that , if $\pi\left(x\right)$ denotes the number of primes less than or equal to $x$, we have $$\pi\left(x\right)\sim\frac{x}{\log x}.$$
In a similar manner ...

**2**

votes

**0**answers

125 views

### Determine whether if $n$ is a primitive pseudoperfect (semiperfect) number, then $\sigma(n)<2^{\sigma_0(n)}$

Determine whether if $n$ is a primitive pseudoperfect (semiperfect) number, then $\sigma(n)<2^{\sigma_0(n)}$.
$\sigma_k(n)$ is the division function and $\sigma(n)=\sigma_1(n)$. A number is ...

**1**

vote

**1**answer

1k views

### Calculating pisano periods for any integer

I recently stumbled across this SPOJ question:
http://www.spoj.com/problems/PISANO/ The question is simple. Calculate the pisano period of a number. After I researched my way through the web, I found ...

**6**

votes

**0**answers

231 views

### Generalizing prime numbers to product-indecomposable objects in toposes

Any object $A$ in any topos decomposes as $A\times1$ and $1\times A$, and in $FinSet$ objects with no other product decompositions are "prime numbers". Is there any extension of the theory of ...

**2**

votes

**4**answers

667 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 ...

**5**

votes

**1**answer

128 views

### Short lattice vectors orthogonal to a random vector

Let $N$ be some prime number.
Suppose I draw $s$ elements $g_1,..., g_s$,
where each $g_i\in [N]$ is taken uniformly from some interval $I_i$ of size, say
$\sqrt{N}$.
Is it possible to provide a ...

**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 ...

**12**

votes

**1**answer

521 views

### Are [Wieferich] primes the only solutions to the equation $2^{k-1} \equiv 1 \pmod{k^2}$?

While studying a certain Diophantine equation in the squarefree integer $k \ge 2$, I believe I have proven the necessary restriction
$$2^{k-1} \equiv 1\!\!\pmod{k^2}. \qquad(\star)$$
Based on what ...

**6**

votes

**1**answer

467 views

### On permuted sum of squares of primes in a list

We want to pick a set of distinct primes (if not possible, then just positive numbers) $p_1,p_2,\dots,p_k$ such that there exists $t$ permutations, ...

**4**

votes

**0**answers

222 views

### Relative Densities

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory.
How can I count ...

**11**

votes

**0**answers

624 views

### Are the twin primes the only positive double zeros of this real function?

Agno's answer
was extremely helpful.
For $x \in \mathbb{R}, x \ge 1$ define
$$ f(x) = \sin\frac{\pi(\Gamma(x)+1)}{\lfloor x \rfloor}$$
By Wilson's theorem the positive integer zeros of $f(x)$ are ...

**5**

votes

**2**answers

324 views

### Function with zeros plus/minus the primes

While playing with Cohen's pari script prodeulerrat found a function.
For $s \in \mathbb{C}$ define
$$ f(s) = \prod_{p \text{ prime}} (1-\frac{s^2}{p^2})$$
The product converges everywhere, no poles ...

**4**

votes

**3**answers

339 views

### Set of primes dividing polynomials and composition

For a non-constant polynomial $A \in \mathbb{Z}[x]$, let $\mathcal{P}(A)$ denote the set of prime numbers $p$ which divide $A(n)$ for some integer $n$. If $\mathcal{P}(A) \subseteq \mathcal{P}(B)$ for ...

**11**

votes

**2**answers

2k views

### An interaction between prime numbers

Let $p_1\ p_2\ \ldots$ be the sequence of all natural prime numbers. There is a slight (just slight) but clear tendency for imitating the number of primes in an interval $(p_k;\ p_n)$ by ...

**1**

vote

**0**answers

309 views

### Green-Tao style theorem for quadratic regressions (Ulam Spiral)

This is a naive question about number theory.
Looking at an Ulam spiral which illustrates primes of the form e.g. $4x^2-2x+c$ and other quadratic equations $ax^2+bx+c$, with $c>0$, there appears a ...

**5**

votes

**1**answer

463 views

### Other implications of Zhang's method

I have been reading a bit about Zhang's proof and the associated Polymath8 project.
Though Tao's high level summary
...

**0**

votes

**0**answers

89 views

### Sequences sharing some primitive prime divisors

Let $q=p^\alpha$ and $q'=p'^\alpha$. Moreover, define $r_i$ and $u_i$ as primitive prime divisor of $q^i-1$ and $q'^i-1$, respectively. Let $\{r_1\}=\{u_1\}$, $\{r_2\}=\{u_2\}$, $\{r_3\}=\{u_3\}$, ...

**5**

votes

**3**answers

714 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 ...

**1**

vote

**0**answers

171 views

### apparent contradiction from result on prime gaps [closed]

I'm looking at Theorem 3 of this paper, which is
If $\:x\geq \exp(\exp(45))\:$ and $\;\;h\:\geq\:3\cdot x^{\frac23}\;\;$ then $$\pi(x+h)-\pi(x) \; \geq \; h\cdot \left(1-\left(3192.34\cdot ...

**0**

votes

**0**answers

112 views

### Composite Euclid Numbers

Is there any way (general procedure, i mean) to determine if a Euclid Number (En = pn# + 1) is prime or composite? Any research papers exploring this theme are also welcome. Thanks!

**8**

votes

**0**answers

127 views

### Does the given operation on pairs of primes always repeat?

Let $p$ and $q$ be two distinct primes. The set
$$A(p,q) =\{m+n : mp+nq=1 \textrm{ and } m,n \in \mathbb{Z}\}$$
is an arithmetic progression. Its step size $p-q$ is coprime to a fixed $m+n$ because
...

**8**

votes

**1**answer

319 views

### How constructive is Dirichlet on primes in progressions?

Is there a known elementary function bound in terms of $a,b,n$ for the $n$-th prime equal to $b$ modulo $a$ (coprime to $b$)?
Bounds on Linnik's constant answer this for the first prime in each ...

**8**

votes

**2**answers

396 views

### Class number of real maximal subfield of cyclotomic fields

Let $p$ be a prime number and $h_p^+$ the class number of $\mathbb{Q}(\zeta_p + \zeta_p^{-1})$. What is known about the values of $p$ for which $h_p^+ = 1$?
Are there infinitely many? Finitely many? ...

**8**

votes

**0**answers

368 views

### Sieve bound for prime $k$-tuples

Let $d_1<d_2<\dots<d_k$ be integers. Then the number of integers $n\leq x$, such that $n+d_1, n+d_2, \ldots, n+d_k$ are simultaneously prime, is bounded above by
$$
\mathfrak{S}(d_1, \ldots, ...

**7**

votes

**1**answer

474 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}
...

**2**

votes

**0**answers

119 views

### Steps required to recognise a $z$-smooth number

I am currently reading section 5 of Pomerance's paper The Number Field Sieve and I have a few questions about smooth numbers.
A number $x\in\mathbb Z_{\ge1}$ is called $z$-smooth if every prime ...

**1**

vote

**0**answers

175 views

### Proof of the Infinitude of Odd Primitive Pseudoperfect Numbers

I'm interested in the infinitude of odd primitive pseudoperfect numbers. Richard K. Guy's book "Unsolved Problems in Number Theory 3rd edition" says that P. Erdős proved the infinitude of odd ...

**1**

vote

**1**answer

84 views

### computing $c_5$ in “Primality testing with Gaussian periods”

As far as I know, the April 2011 version of #143 on this page has not been improved upon.
On page 10 of that paper, the authors give an algorithm that uses a constant $\:c_{\hspace{.01 in}5}\:$.
...

**10**

votes

**1**answer

410 views

### Least prime $p$ such that an irreducible polynomial of degree $n$ has no root modulo $p$?

This question is inspired by an old question of Greg Kuperberg, about how small is the first prime $p$ which makes a given monic polynomial $P$ with integral coefficient have a (simple) root modulo ...

**2**

votes

**0**answers

143 views

### Best known Upper bound on Twin Primes [duplicate]

I know that there is a result from J Wu that the number of twin primes less than a given magnitude $N$ does not exceed
$$\frac{2aCN}{\log^2{N}}$$
Where $C=\prod \frac{p(p-2)}{(p-1)^2}$ and $a$ is ...

**33**

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 ...

**17**

votes

**1**answer

788 views

### Infinitely many primes, and Mobius randomness in sparse sets

Problem 1: Find a (not extremely artificial) set A of integers so that for every $n$, $|A\cap [n]| \le n^{0.499}$, ($[n]=\{1,2,...,n\}$,) where you can prove that $A$ contains infinitely many primes.
...

**2**

votes

**1**answer

522 views

### Prime numbers of a special kind

Is it proved that there is only a finite number of prime numbers of the form $2^{2n} + 2^n + 1$ (where $n \in \mathbb{N}$)?

**10**

votes

**1**answer

524 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 ...

**5**

votes

**2**answers

446 views

### General Euclid-Fermat sequences

Question
Fermat sequence
$$ \forall_{n=1\ 2\ \ldots}\quad F_n:=2^{2^{n-1}}+1$$
has every two different terms relatively prime, and $1$ does not appear as any of them. Thus it would be ...

**12**

votes

**2**answers

735 views

### An Euler-proof that cannot be repaired?

Ed Sandifer writes on Euler's wonderful work on prime numbers: The sum of the series of reciprocals of prime numbers $\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$ is infinitely large, and is infinitely ...

**8**

votes

**1**answer

623 views

### Density of prime pairs whose gap is less than the average gap

By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap ...

**1**

vote

**0**answers

131 views

### Division between deformed Gausian integers

Let $P_3=${$z\in Z[\sqrt{-3}],|z|^2 \text { is a prime number}, >3$}
Let $\alpha_1,...\alpha_n$ be distinct elements in $P_3$, and $l_1,...l_n\in Z^+$. Set ...

**2**

votes

**0**answers

321 views

### On the primitive prime divisors of $q^n-1$

Let $q=p^\alpha$ be a prime power.
We call $r$ a primitive prime divisor of $q^n-1$ where $r\mid (q^n-1)$ but $r\nmid (q^i-1)$ for each $1\leq i\leq n-1$. The set of all primitive prime divisors of ...

**1**

vote

**1**answer

92 views

### On the set of divisors of $q-1$ and $q'-1$

For a natural number $n$ we denote by $\pi(n)$, for the set of prime divisors of $n$.
Let $q=p^\alpha$ and $q'=r^\beta$ where $r$ and $p$ are odd prime numbers and $\alpha,\beta$ are natural numbers.
...

**0**

votes

**1**answer

226 views

### For any n and some prime p there is an elemnet in Zp* of order n [closed]

How can I prove, that for any positive integer $n>0$ there is a prime $p$, such that the multiplicative group of the residue ring $Z_p^*$ contains an element $a$ of order $n$? No ideas at all...

**6**

votes

**0**answers

610 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, ...

**-1**

votes

**1**answer

348 views

### Mersenne Prime Sequences

Hi.
Given the following sequence (of Mersenne primes):
$ A_{1} = 2 $
$ A_{n} = 2^{A_{n-1}} - 1 $
The first five elements are all prime numbers:
$ 2 $
$ 2^{2}-1=3 $
$ 2^{3}-1=7 $
$ 2^{7}-1=127 ...

**2**

votes

**2**answers

832 views

### Finding primes using Euler's sum of divisors recurrence relation

Euler came up with following recurrence relation for the sum of divisors (refer to http://arxiv.org/abs/math/0411587)
$$\sigma(n) = \sigma(n−1) + \sigma(n−2) − \sigma(n−5) − \sigma(n−7) \dots$$
Since ...

**3**

votes

**1**answer

339 views

### The Nth number with M prime factors

Hi.
Suppose we arrange all natural numbers in a matrix P defined as follows:
P[I][J] = The Jth number with I prime factors. So P looks something like:
1
2 , 3 , 5 , 7 , 11 , 13 , 17 ...

**0**

votes

**0**answers

138 views

### Prime-Counting Function

Would a summation of floor[cos^2π(((j-1)!+1)/j)] from j=1 to x be the same as π(x)?

**1**

vote

**1**answer

162 views

### Behavior of a quantity related to Fermat's 4n + 1 Theorem

One of Fermat's theorems states that if $p = 4n + 1$ for some integer $n$, then $p$ can be expressed uniquely as a sum of two squares, $p = a^2 + b^2$. I am working on a problem and I would like to ...

**3**

votes

**1**answer

258 views

### Primes in short intervals with a preassigned frobenius

Edited after mistake in the first version.
It is known since Selberg that under the Riemann Hypothesis, given an $\epsilon>0$, there is a prime between $x$ and $x+O(x^\epsilon)$ for all $x$ in a ...