Related papers: On Grosswald's conjecture on primitive roots

The Cram\'er-Granville conjecture is an upper bound on prime gaps, $g_n = p_{n+1} - p_n < \cCramer \, \log^2 p_n$ for some constant $\cCramer \geq 1$. Using a formula of Selberg, we first prove the weaker summed version: $\sum_{n=1}^N g_n <…

Given a prime $p\geq 5$, we reduce modulo p a convolution of order p-1 of powers of two weighted Bernoulli numbers with Bernoulli numbers in terms of harmonic numbers and generalized harmonic numbers. Our proof is based on studying the…

Let $p\equiv 1\,(\mathrm{mod}\,3)$ be a prime and denote by $\zeta_3$ a primitive third root of unity. Recently, Lemmermeyer presented a conjecture about $3$-class groups of pure cubic fields $L=\mathbb{Q}(\sqrt[3]{p})$ and of their normal…

For a number field $k$ and an odd prime $p$, let $\tilde{k}$ be the compositum of all the ${\mathbb Z}_p$-extensions of $k$, $\tilde{\Lambda }$ the associated Iwasawa algebra, and $X(\tilde{k})$ the Galois group over $\tilde{k}$ of the…

For a prime number $p$ and a number field $k$, let $\tilde{k}$ be the compositum of all $\mathbb{Z}_p$-extensions of $k$. Greenberg's Generalized Conjecture (GGC) claims the pseudo-nullity of the unramified Iwasawa module $X(\tilde{k})$ of…

We prove that any multiplicative subgroup G of the prime field f_p with |G| < p^{1/2} satisfies |3G| \gg |G|^2 / \log |G|. Also, we obtain a bound for the multiplicative energy of any nonzero shift of G, namely E^* (G+x) \ll |G|^2 log |G|,…

Mathematicians has been trying to prove the weak Goldbach's conjecture by adding prime numbers, as stated in the conjecture. However, we believe that the solution does not need to be analytically solved. Instead of trying to add prime…

We study the difference between the number of primitive roots modulo $p$ and modulo $p+k$ for prime pairs $p,p+k$. Assuming the Bateman-Horn conjecture, we prove the existence of strong sign biases for such pairs. More importantly, we prove…

Let A be an abelian variety over a number field k. We show that weak approximation holds in the Weil-Ch\^atelet group of A/k but that it may fail when one restricts to the n-torsion subgroup. This failure is however relatively mild; we show…

The Elementary Type Conjecture in Galois theory provides a concrete inductive description of the finitely generated maximal pro-$p$ Galois groups $G_F(p)$ of fields $F$ containing a root of unity of order $p$. We describe several variants…

Building on earlier papers of several authors, we establish that there exists a universal constant $c > 0$ such that the minimal base size $b(G)$ of a primitive permutation group $G$ of degree $n$ satisfies $\log |G| / \log n \leq b(G) < 45…

In this paper two conjectures are proposed based on which we can prove the first case of Fermat's Last Theorem(FLT) for all primes $p \equiv -1 (\bmod~6)$. With Pollaczek's result {\bf [1]} and the conjectures the first case of FLT can be…

For a primitive Hecke-Maass cusp form $\phi$ of level $N$ with the $n$-th Hecke eigenvalue $\lambda_{\phi}(n)$ and a prime number $p\nmid N$, the celebrated Ramanujan conjecture at $p$ asserts the following sharp upper bound: \[…

This paper establishes an extension of Wolstenholme's theorem to the ring of Gaussian integers $\mathbb{Z}[i]$. For a prime $p > 7$, we prove that the sum $S_p$ of inverses of Gaussian integers in the set $\{n+mi \mid 1 \leq n, m \leq p-1,…

Let $G$ be a finite group and $\mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. Quillen conjectured that $O_p(G)$ is nontrivial if $\mathcal{A}_p(G)$ is contractible. We prove that $O_p(G)\neq 1$ for any…

For a fixed integer $n\geq 2,$ we show that a permutation of the least residues mod $p$ of the form $f(x)=Ax^k$ mod $p$ cannot map a residue class mod $n$ to just one residue class mod $n$ once $p$ is sufficiently large, other than the maps…

Let $p$ be a fixed prime number, and $N$ be a large integer. The 'Inverse Conjecture for the Gowers norm' states that if the "$d$-th Gowers norm" of a function $f:\F_p^N \to \F_p$ is non-negligible, that is larger than a constant…

Every finite group whose order is divisible by a prime $p$ has at least $2 \sqrt{p-1}$ conjugacy classes.

Let $G$ be a semisimple algebraic group over a field of characteristic $p > 0$. We prove that the dual Weyl modules for $G$ all have $p$-filtrations when $p$ is not too small. Moreover, we give applications of this theorem to…

For a prime $p$ and an integer $u$ with $\gcd(u,p)=1$, we define Fermat quotients by the conditions $$ q_p(u) \equiv \frac{u^{p-1} -1}{p} \pmod p, \qquad 0 \le q_p(u) \le p-1. $$ D. R. Heath-Brown has given a bound of exponential sums with…