**1**

vote

**3**answers

326 views

### Has this formula for $G_{k}:=\lim\inf_{n\to\infty}p_{n+k}-p_{n}$ been conjectured?

I give here a heuristics that suggests that the quantity $\displaystyle{G_{k}:=\liminf_{n\to\infty}p_{n+k}-p_{n}}$ should be approximately equal to $k(1+H_{k})$, where $H_{k}$ is the $k$-th harmonic ...

**12**

votes

**0**answers

538 views

### Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system,
$$x_1^2+x_2^2+\dots+x_n^2 = y^2$$
$$x_1^3+x_2^3+\dots+x_n^3 = z^3$$
and define the function,
$$F(s_m) = x_1+x_2+\dots+x_n$$
For $n\geq3$, using an ...

**6**

votes

**1**answer

437 views

### Are reduced residue systems relative primorials an active area of research? If not, why not?

As a math amateur, I am finding the study reduced residue systems relative a primorial a very interesting way to understand the distribution of primes. For example, it is fascinating to me that it is ...

**3**

votes

**4**answers

263 views

### Uniform upper bound for the sum over primes $\sum_{p \leq x} p^{-1+\varepsilon}$

I am reading the article D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprime, Math. Comp. (1991) (click).
In equation (27) the authors, apparently, used the following ...

**8**

votes

**1**answer

278 views

### Asymptotic limit of truncated Legendre sieve

Consider the truncated sum
$$
S(x):=\sum_{\substack{{d\mid P(\sqrt{x})}\\{d\leq x}}}\mu(d)/d,
$$
where $P(z)$ is the product of all primes less than or equal to $z$, and $\mu(d)$ is the Möbius ...

**19**

votes

**3**answers

1k views

### A Polynomial With Positive Prime Density

Let $P(x)$ be a non-constant polynomial with real coefficients.
Can natural density of
$$\{n\ |\ \lfloor P(n)\rfloor \ \text{is prime.}\}$$
be positive?

**2**

votes

**0**answers

151 views

### Squarefree part of a Mersenne number

Consider the Mersenne number; $M_p=2^p−1$.
Let $M_p=a_pb^2_p$ where $a_p$ is positive, squarefree, and $p$ is prime.
A chinese paper written by Le Maohua "“On Mersenne Numbers”" states that the ...

**6**

votes

**1**answer

307 views

### Primes isolated by large gaps to either side

Say that the $n$-th prime $p_n$ is isolated to degree $k$
(my notation) if
the prime gap to either side is larger than $\log p_n$ to the $k$-th power:
\begin{eqnarray*}
p_n - p_{n-1} & > & ...

**2**

votes

**2**answers

361 views

### Primes as uncorrelated random variables [closed]

The heuristic justification section of the Wikipedia article about Goldbach's conjecture says that the argument that suggests that
the number of twin primes below $x$ should be roughly ...

**14**

votes

**3**answers

2k views

### A variant of Goldbach Conjecture

I'm asking if this variant of weak Goldbach's Conjecture is already known.
Let $N$ be an odd number. Does there exists prime numbers $p_1$, $p_2$ and $p_3$ such that $p_1+p_2-p_3=N$? Ideally, can we ...

**17**

votes

**2**answers

551 views

+100

### Small quotients of smooth numbers

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...

**8**

votes

**2**answers

939 views

### divisible by all standard prime numbers

This question is about prime numbers in nonstandard models of Peano Arithmetic. Every such model looks like N+AxZ, where A is a dense linear order without end points.
There are many nonstandard ...

**6**

votes

**0**answers

256 views

### Can integers be distorted to make primes more regular?

Given a set $P$ of real numbers $\ge 1$, define the gap among different products in $P$ as
$$g(P) = \inf \big\{\prod_{i=1}^n p_i^{a_i} - \prod_{i=1}^n p_i^{b_i} \mid p_i\in P;\,\, p_i\ne p_j \,\text{ ...

**1**

vote

**0**answers

137 views

### Generalization of Little Fermat Theorem for a particular $a$ and perfect shuffles

I'm looking for the smallest $n\in \mathbb{N}$ that solves the following equation:
$$2^n=1 \mod m$$
For an odd $m$. I know that Little Fermat Theorem and Euler Totient give me a solution but they ...

**3**

votes

**1**answer

115 views

### What is the Complexity Class of the “Function Variant” of the Integer Factorization Problem?

I've been reading up a lot Prime Factorization and it's complexity, including a fair number of questions on this very site. However, I still feel there is a question still left unanswered.
So, ...

**2**

votes

**0**answers

138 views

### Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula?

My question is on Mobius inversion formula convergence/properties when used with infinite sums of function.
Lets consider (on $\mathbb{R}^{+}$):
$$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$
We call ...

**2**

votes

**1**answer

329 views

### Can you find squares in this class?

For a problem I am working over, I would like to prove that numbers of the following type are not squares
$p(l^4+6l^2m^2-3m^4)$
where $p,l,m$ are integers an $p$ prime. I have already found various ...

**7**

votes

**1**answer

428 views

### Which kind of subsets of primes one needs to generate a positive ratio of the natural numbers?

Not knowing elementary number theory well, I ask this one, which is not very clear to answer, rather I am looking for some results around this question or known theorems. The problem is the following:
...

**0**

votes

**0**answers

105 views

### Polynomial congruences with respect to a large prime (power)

Suppose that $f(x) \in \mathbb{Z}[x]$ is a polynomial of degree $d$ such that for all primes $p$, there exists $x_0 \in \mathbb{N}$ such that $p^2 \nmid f(x_0)$. Further, suppose that $f$ is ...

**5**

votes

**1**answer

369 views

### A set of integers whose factorial can be written as a product of two factorials

I am trying to collect informations concerning the set
$$\mathcal{A}=\left\{n\in\mathbb{N} \mid (\exists k,l\in\{2,3,\dots,n-2\})(n!=k!l!)\right\}.$$
It seems not much is known about the set ...

**-3**

votes

**1**answer

302 views

### Asymptotic formula for $\prod_{p\leq x} (1-p^{-1})$ [closed]

Does there exists a good asymptotic formula for
$$A(x) := \prod_{p\leq x}(1-\frac 1p).$$
By using a heuristic argument one can guess:
$$A(x) \sim \frac{1}{2\,\mathrm{ln}(x)}.$$
Here is the ...

**1**

vote

**0**answers

182 views

### Density of numbers whose prime factors all come from a fixed congruence class

Let $q$ be a positive integer greater than one, and let $a$ be an integer such that $\gcd(a,q) = 1$. Define
$$D(a,q) = \{n \in \mathbb{N} : p | n \Rightarrow p \equiv a \pmod{q} \}.$$
Do we know the ...

**2**

votes

**1**answer

410 views

### Is anything like $\phi(n)>\dfrac n{e^\gamma\log\log n},\ \sigma(n)<e^\gamma n\log\log n$ known/conjectured for the generalizations of these functions?

Is anything like $\dfrac n{\phi(n)}<\dfrac{\sigma(n)}n<e^\gamma\log\log n$ known/conjectured for the generalizations of these functions?
Let $n=p_1^{a_1}\cdots p_t^{a_t}$ be the canonical prime ...

**10**

votes

**2**answers

723 views

### Most dense subset of numbers that avoids arbitrarily long arithmetic progressions

The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression.
I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid
...

**2**

votes

**0**answers

101 views

### Best constant for Maier's theorem?

Maier proved that, for fixed $\lambda>1,$
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\pi(x)}{\log^{\lambda-1}x}>1
$$
and in particular
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda ...

**8**

votes

**2**answers

834 views

### What is known about primes of the form x^2-2y^2?

David Cox's book Primes of The Form: $x^2+ny^2$ does a great job proving and motivating a lot of results for $n>0$. I was unable to find anything for negative numbers, let alone the case I am ...

**2**

votes

**1**answer

278 views

### $n^3 | \sum_{i=1}^{n-1}\binom{n}{i}^2$ => $n | \sum_{i=1}^{n-1}\binom{n}{i}$?

For $n\in \mathbf{N}$ is $$n^3 \text{ divides } \sum_{i=1}^{n-1}\binom{n}{i}^2=\binom{n}{1}^2+\cdots +\binom{n}{n-1}^2$$ impling $$n \text{ divides } ...

**0**

votes

**0**answers

95 views

### logarithmic integral question

Define: $\operatorname{li}(x)=\int_{0}^{x}\dfrac{1}{\log(t)}\operatorname{d}t$.
When does the following statement fail?
With $\theta = 1 + \frac{1}{\operatorname{li}(x)}$, for $x \ge x_0$,
...

**0**

votes

**0**answers

198 views

### Does the Euler product converge at $s=1$ for the Dirichlet $L$ function?

For the Riemann Zeta function, the Euler product converges on $\{Re(s)=1\}$ except at $s=1$.The zeta series diverges everywhere on $\{Re(s)=1\}$. But the $L$ series converges on $\{Re(s)>0\}$. What ...

**16**

votes

**1**answer

780 views

### Primes that are sums of two squares with constraints on the squares

It is well known that there are infinitely many primes of the form $a^2+b^2$ (namely all primes congruent to $1$ modulo $4$). On the other hand, Euler raised the problem as to whether there are ...

**5**

votes

**1**answer

390 views

### Generating primes with floor of a polynomial $[p(n)]$

Is there a polynomial $p(x)$ with real coefitients and degree at least one that $[p(n)]$ for everey natural number like $n$ be a prime?
If yes, what is such a polynomial $p(x)$ and if no, how to ...

**12**

votes

**0**answers

324 views

### Word complexity of primes mod 4

For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...

**4**

votes

**0**answers

228 views

### Analog of Euler's factoring technique

Is there an analog of Euler's Two Squares factoring theorem over polynomial rings $\Bbb Z[x]$ by considering a version for non-negative polynomials?
Euler's two squares factoring states that numbers ...

**5**

votes

**2**answers

875 views

### How to calculate the sum of remainders of N?

I'm trying to sum the remainders when dividing N by numbers from 1 up to N
$$\sum_{i = 1}^{N} N \bmod i$$
It's easy to write a program to evaluate the sum if N is small in O(N) but what if N is large ...

**4**

votes

**0**answers

332 views

### Is there a hidden symmetry in the prime numbers distribution?

Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime.
...

**1**

vote

**2**answers

623 views

### Has this formula about prime gaps already been conjectured and/or proven?

While playing around with prime gaps, I found out that the following formula seems to be a rather good approximation of the ratio $\dfrac{p_{b}-p_{a}}{b-a}$ where $a<b$ are positive integers:
...

**1**

vote

**0**answers

461 views

### Is $ a^2b^2-ab-ap$ a perfect square for suitable $a,b\in \mathbb{Z}^+$?

Consider the expression
$$
a^2b^2-ab-ap\qquad (a,b\in \mathbb{Z}^+),
$$
where $p\equiv1\pmod{4}$.
Question. For every prime $p\equiv1\pmod{4}$ do there exist $a,b\in\mathbb{Z}^+$ such that ...

**2**

votes

**1**answer

269 views

### Prime Number Theorem on APs under various conjectures

I'm trying to find the best asymptotic expansions for $\pi(x; a, q)$ in various states:
Unconditionally we have
\begin{equation}
\pi(x; a, q) = \frac{\operatorname{li(x)}}{\phi(q)} + O\left(x ...

**2**

votes

**0**answers

421 views

### A “Take a Square Root When You Can” conjecture related to the prime factorization

I would tend to think that the following has already been investigated.
But as implied from the title, I have no idea how to even start looking for it.
Let $P_n$ denote the sum of the squares of ...

**1**

vote

**0**answers

58 views

### closed form for a series with binomials and primes

does the series $\sum_{n=0}^\infty p^n \binom{x}{p^n}$ have a closed form ? ($p$ prime)
this is a special case of $\sum_{n=0}^\infty p^n \left(\sum_{k=p^n}^{p^{n+1}-1}a_k\binom{x}{k}\right)$ with the ...

**2**

votes

**1**answer

723 views

### Conjecture on the square root of the sum of the squares of the prime factors of a number

Let $A_{n}$ denote the square root of the sum of the squares of the prime factors of $n$.
For example, $A_{60}=\sqrt{2^2+2^2+3^2+5^2}\approx6.48$.
I have recently made the following observations:
...

**10**

votes

**1**answer

660 views

### Divergence of a series similar to $\sum\frac{1}{p}$

Suppose we start with $k$ primes $p_1,p_2,\ldots ,p_k$ (not necessarily consecutive) and a residue class for each prime $r_1,r_2,\ldots ,r_k$.
We denote the least integer not covered by the arithmetic ...

**8**

votes

**3**answers

621 views

### Is $n = p-q$ equivalent to Goldbach's Conjecture?

One open conjecture is that every even integer greater than two is the difference of two primes. (Some superficial discussion here.)
Goldbach's conjecture states that every even integer greater than ...

**5**

votes

**1**answer

614 views

### Any way to prove Prime Number Theorem using Hyperbolic Geometry? [closed]

The prime number theorem says that the density of prime numbers is inverse as the number of digits of $n$:
$$\displaystyle \frac{\{1 \leq k \leq n : \text{ prime } \}}{n} \approx \frac{1}{\log n}$$
...

**2**

votes

**2**answers

457 views

### Prime divisors of $p^n+1$

Let $p$ be a rational prime and $n$ be a positive integer.
It can be easily deduced from Zsigmondy's theorem that $p^n+1$ has a prime divisor greater than $2n$ except when $(p,n)=(2,3)$ or ...

**5**

votes

**1**answer

384 views

### Ruth-Aaron triples, etc

A Ruth-Aaron pair is two numbers $(n,n+1)$ such that
their sum of prime factors is equal, counting repeated prime factors.
(The name refers to Hank Aaron's 715 homeruns surpassing Babe Ruth's 714!)
So
...

**3**

votes

**0**answers

452 views

### Two (strictly related) proofs by induction of inequalities

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...

**3**

votes

**1**answer

306 views

### Primes $p=x^2+27y^2$ and Ramanujan's $x_1^{1/3} + x_2^{1/3} + x_3^{1/3}$

I was trying to generalize,
...

**6**

votes

**1**answer

369 views

### Prime labelling of graphs

A prime labeling of a graph is an injective function $f: V(G) \to \{1, 2, ..., |V(G)|\}$
such that for every pair of adjacent vertices $u$ and $v$, $\text{gcd}(f(u), f(v)) = 1$ (labels of any two ...

**2**

votes

**1**answer

147 views

### Cardinality of the prime divisor set of a k-power sum

Let $a_{1},\dots,a_{n}$ be positive natural numbers ($n>2$) such that $a_{i}\neq a_{j}$ if $i\neq j$. I want to prove that
$$ \left\lvert \left\{ p \text{ prime} \; : \; p \mid \sum_{i=1}^n ...