Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a ...

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8
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0answers
78 views

The multiplicative group generated by shifted primes

I am asking for references about the following problem. In particular, it is still open? If not, what is the state of the art result? Problem 1. Let $\Gamma$ be the multiplicative subgroup of ...
-1
votes
0answers
147 views

A not-so-weak Goldbach's conjecture

While Goldbach's conjecture (every even integer greater than 2 can be expressed as the sum of two primes) remains open, one can weaken the question by asking whether every (even,odd) integer can be ...
40
votes
4answers
1k views

How did Cole factor $2^{67}-1$ in 1903

I just heard a This American Life episode which recounted the famous anecdote about Frank Nelson Cole factoring $N:=2^{67}-1$ as $193707721 \times 761838257287$. There doesn't seem to be a historical ...
6
votes
0answers
62 views

Prime divisors of the norm of the first coefficient of an elliptic newform at width-1 cusps.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ and let $f$ be its newform. Suppose $p \geq 5$ is a prime such that $p^2 \mid N$. We assume $f$ is $p$-minimal, which is equivalent to that the ...
3
votes
2answers
353 views

Non-standard Gauss sums

I have the following problem. Let $p$ be some prime. What is the value of \begin{equation} \sum_{k=1}^{p-1} \left(\frac{k+1}{p}\right) \omega_p^{kl}, \end{equation} where $\left(\frac{k+1}{p}\right)$ ...
0
votes
2answers
157 views

Smallest constant so that there are at least $n/\log_2{n}$ primes between $n$ and a constant multiple of $n$

What is the smallest known $c$ so that for any $n\geq 2$ there are at least $n/\log_2{n}$ primes between $n$ and $cn$ (inclusive)? The prime number theorem seems to give an asymptotic result so I am ...
0
votes
0answers
68 views

Expliciting the distance between consecutive Goldbach numbers assuming it's finite

In this paper, the author shows unconditionally that at least one of the following statements holds: i) the distance between two consecutive Goldbach numbers is finite, i.e. there exists an ...
1
vote
0answers
45 views

Asymptotics on number of bounded prime gaps [duplicate]

It's been over 2 years since the groundbreaking paper by Yitang Zhang in which he has shown that infinitely many prime pairs are by some constant $H$, with $H\leq 70000000$. Over the course of the ...
0
votes
0answers
86 views

A question on the Siegel's theorem in specific condition

I want to know the number of expressions such that \begin{align} x=p+aq \end{align} for sufficiently large even number $x$, where $p$ and $q$ are prime numbers and $a$ is a positive odd integer which ...
4
votes
0answers
246 views

Are there always at least *five* divisions?

@JosephO'Rourke asked a question about a Collatz like function related to primes: $f(n) = \begin{cases} n^2 & \text{if} \;n \;\text{is prime} \\ \lfloor n/2 \rfloor & \text{if} \;n ...
3
votes
3answers
373 views

Are There Infinitly Many $n$ Which $a\times n!+1$ Be Composite?

$ax+1$ is a linear polynomial with integral coefficients. Are there infinitly many $n$ which $a\times n!+1$ be composite? As I know this problem is true for polynomials with degree greater that 1, ...
13
votes
1answer
443 views

A Collatz-like function that bifurcates on primes

This is likely piling one mystery on another, but ... I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows: $$ f(n) = \begin{cases} n^2 & \text{if} \;n \;\text{is ...
1
vote
4answers
586 views

Distribution of composite numbers

I have moved this question to math.stackexchange.com. People who are interested in this question can discuss at :http://math.stackexchange.com/questions/1272431/distribution-of-composite-numbers ...
3
votes
0answers
105 views

Farey Fractions Estimate Equivalent to the Prime Number Theorem?

Wikipedia's article on Farey Fractions points to an article of Jerome Franel that some averages are equivalent to the Riemann hypothesis. Let $F_n$ be the $n$-th Farey sequence, then the number of ...
0
votes
1answer
163 views

Is the following conjecture equivalent to the Second Hardy-Littlewood Conjecture? [closed]

Let $y$ be an arbitrary positive real number such that $y\ge 2$. Then if we can prove that, $$\lim_{x\to\infty}\dfrac{\pi(x)+\pi(y)}{\pi(x+y)}=1$$will it imply that for all sufficiently large $x$ ...
1
vote
3answers
295 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 ...
10
votes
0answers
409 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
1answer
370 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
4answers
221 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
1answer
243 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 ...
18
votes
3answers
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
0answers
142 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
1answer
288 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
2answers
341 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
3answers
1k 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 ...
11
votes
1answer
283 views

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
2answers
854 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
0answers
238 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
0answers
130 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 ...
-4
votes
2answers
305 views

Brocard's problem [closed]

According to Brocard's problem $$x^{2}-1=n!=4!*(5+1)(5+2)...(5+s)$$ here,$(5+1)(5+2)...(5+s)=\mathcal{O}(5^{r}),4!=k$. So, $$x^{2}-1=k *\mathcal{O}(5^{r})$$ Here, $\mathcal{O}$ is Big O ...
3
votes
1answer
91 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
0answers
116 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
1answer
326 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
1answer
380 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
0answers
86 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
1answer
301 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
1answer
285 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
0answers
160 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
1answer
333 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 ...
7
votes
2answers
570 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
0answers
91 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
2answers
717 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
1answer
270 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
0answers
89 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
0answers
161 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
1answer
765 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
1answer
368 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
0answers
310 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
0answers
215 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 ...
4
votes
2answers
669 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 ...