# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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} & > & ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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Given a set $P$ of real numbers $\ge 1$, define the gap among different products in $P$ as $$g(P) = \inf \big\{\prod_{i=1}^n p_i^{a_i} - \prod_{i=1}^n p_i^{b_i} \mid p_i\in P;\,\, p_i\ne p_j \,\text{ ... 0answers 137 views ### Generalization of Little Fermat Theorem for a particular a and perfect shuffles I'm looking for the smallest n\in \mathbb{N} that solves the following equation:$$2^n=1 \mod m$$For an odd m. I know that Little Fermat Theorem and Euler Totient give me a solution but they ... 1answer 115 views ### What is the Complexity Class of the “Function Variant” of the Integer Factorization Problem? I've been reading up a lot Prime Factorization and it's complexity, including a fair number of questions on this very site. However, I still feel there is a question still left unanswered. So, ... 0answers 138 views ### Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula? My question is on Mobius inversion formula convergence/properties when used with infinite sums of function. Lets consider (on \mathbb{R}^{+}):$$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$We call ... 1answer 329 views ### Can you find squares in this class? For a problem I am working over, I would like to prove that numbers of the following type are not squares p(l^4+6l^2m^2-3m^4) where p,l,m are integers an p prime. I have already found various ... 1answer 428 views ### Which kind of subsets of primes one needs to generate a positive ratio of the natural numbers? Not knowing elementary number theory well, I ask this one, which is not very clear to answer, rather I am looking for some results around this question or known theorems. The problem is the following: ... 0answers 105 views ### Polynomial congruences with respect to a large prime (power) Suppose that f(x) \in \mathbb{Z}[x] is a polynomial of degree d such that for all primes p, there exists x_0 \in \mathbb{N} such that p^2 \nmid f(x_0). Further, suppose that f is ... 1answer 369 views ### A set of integers whose factorial can be written as a product of two factorials I am trying to collect informations concerning the set$$\mathcal{A}=\left\{n\in\mathbb{N} \mid (\exists k,l\in\{2,3,\dots,n-2\})(n!=k!l!)\right\}.$$It seems not much is known about the set ... 1answer 302 views ### Asymptotic formula for \prod_{p\leq x} (1-p^{-1}) [closed] Does there exists a good asymptotic formula for$$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)}.$$Here is the ... 0answers 182 views ### Density of numbers whose prime factors all come from a fixed congruence class Let q be a positive integer greater than one, and let a be an integer such that \gcd(a,q) = 1. Define$$D(a,q) = \{n \in \mathbb{N} : p | n \Rightarrow p \equiv a \pmod{q} \}.$$Do we know the ... 1answer 410 views ### Is anything like \phi(n)>\dfrac n{e^\gamma\log\log n},\ \sigma(n)<e^\gamma n\log\log n known/conjectured for the generalizations of these functions? Is anything like \dfrac n{\phi(n)}<\dfrac{\sigma(n)}n<e^\gamma\log\log n known/conjectured for the generalizations of these functions? Let n=p_1^{a_1}\cdots p_t^{a_t} be the canonical prime ... 2answers 723 views ### Most dense subset of numbers that avoids arbitrarily long arithmetic progressions The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression. I am wondering: How dense can a subset S \subset \mathbb{N} be and still avoid ... 0answers 101 views ### Best constant for Maier's theorem? Maier proved that, for fixed \lambda>1,$$ \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 ...
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 ...