Related papers: Bubbles of congruent primes

A conjecture of Cai-Zhang-Shen for figurate primes says that every integer $k>1$ is the sum of two figurate primes. In this paper we give an equivalent proposition to the conjecture. By considering extreme value problems with constraints…

We show that the semi-simplicity conjecture for finitely generated fields follows from the conjunction of the semi-simplicity conjecture for finite fields and for the maximal abelian extension of the field of rational numbers.

For a given positive integer $k$, we prove that there are at least $x^{1/2-o(1)}$ integers $d\leq x$ such that the real quadratic fields $\mathbb Q(\sqrt{d+1}),\dots,\mathbb Q(\sqrt{d+k})$ have class numbers essentially as large as…

Let $m$ be a natural number, and let $\mathcal{Q}$ be a set containing at least $\exp(C m)$ primes. We show that one can find infinitely many strings of $m$ consecutive primes each of which has some $q\in\mathcal{Q}$ as a primitive root,…

Let $\mathbb{Q}(\alpha)$ and $\mathbb{Q}(\beta)$ be linearly disjoint number fields and let $\mathbb{Q}(\theta)$ be their compositum. We prove that the first-degree prime ideals of $\mathbb{Z}[\theta]$ may almost always be constructed in…

We give an explicit form of Ingham's Theorem on primes in the short intervals, and show that there is at least one prime between every two consecutive cubes $x\sp{3}$ and $(x+1)\sp{3}$ if $\log\log x\ge 15$.

In terms of the creative microscoping method recently introduced by Guo and Zudilin [Adv. Math. 346 (2019), 329--358], we find a $q$-supercongruence with four parameters modulo $\Phi_n(q)(1-aq^n)(a-q^n)$, where $\Phi_n(q)$ denotes the…

We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.

The results for the fractional sequence $\left \{[x/n]+1:n \leq x\right \}$, and the fractional sequence in arithmetic progression $\left \{q[x/n]+a:n \leq x\right \}$, where $a<q$ are integers such that $\gcd(a,q)=1$, prove that these…

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the…

The fifteen supersingular primes, see https://oeis.org/A002267, appear in the theory of the moduli of abelian surfaces. This short expository note explains why.

We prove that for any sequence of binary alphabets $\mathcal{A}_1,\mathcal{A}_2,\dots$, there exists a cube-free word $c_1c_2\dots$ so that $c_1\in\mathcal{A}_1,c_2\in\mathcal{A}_2,\dots$. In particular, for every $n$, there are at least…

We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidability. We also obtain a…

In this paper we study the problem of detecting prime numbers between all consecutive cubes. Firstly, we use a large computation to show that there is always a prime between $n^3$ and $(n+1)^3$ for $n^3\leq 1.649\cdot 10^{40}$. In addition,…

Let $s$ be a fixed positive integer constant, $\varepsilon$ be a fixed small positive number. Then, provided that a prime $p$ is large enough, we prove that for any set $\{{\mathcal M}\subseteq \mathbb F_p^*$ of size $|{\mathcal M}|=…

This paper proposes an elementary solution to a special case of finding all perfect squares that can be written as sum of consecutive integer cubes. It is shown that there are no non-trivial solutions if the perfect square is a prime power,…

For any $\epsilon>0$, there exists $q_0(\epsilon)$ such for any $q\ge q_0(\epsilon)$ and any invertible residue class $a$ modulo $q$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly three…

Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: The primes in an short interval contains many arithmetic progressions of any…

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…

We prove that, if $m,n\geqslant 1$ and $a_1,\ldots,a_m$ are nonnegative integers, then \begin{align*} \frac{[a_1+\cdots+a_m+1]!}{[a_1]!\ldots[a_m]!}\sum^{n-1}_{h=0}q^h\prod_{i=1}^m{h\brack a_i} \equiv 0\pmod{[n]}, \end{align*} where…