Related papers: On Class Numbers of Pure Quartic fields

For a prime $p$ and a rational elliptic curve $E_{/\mathbb{Q}}$, set $K=\mathbb{Q}(E[p])$ to denote the torsion field generated by $E[p]:=\operatorname{ker}\{E\xrightarrow{p} E\}$. The class group $\operatorname{Cl}_K$ is a module over…

Following the natural instinct that when a group operates on a number field then every term in the class number formula should factorize `compatibly' according to the representation theory (both complex and modular) of the group, we are led…

Let $K$ be a totally real number field. For all prime number $p\geq 5$, let us denote by $F_p$ the Fermat curve of equation $x^p+y^p+z^p=0$. Under the assumption that $2$ is totally ramified in $K$, we establish some results about the set…

Let p be an odd prime. Let K_p = Q(zeta) be the p-cyclotomic field. Let pi be the prime ideal of K_p lying over p. Let G be the Galois group of K_p. Let v be a primitive root mod p. Let sigma be a Q-isomorphism of K_p. Let P(sigma) =…

Given a pair of regular quadratic forms over $\mathbb{Q}$ which are in the same genus and a finite set of primes $P$, we show that there is an effective way to determine a rational equivalence between these two quadratic forms which are…

Let $\ell>3$ be a prime such that $\ell \equiv 3 \pmod{4}$ and $\mathbb{Q}(\sqrt{\ell})$ has class number 1. Then Hirzebruch and Zagier noticed that the class number of $\mathbb{Q}(\sqrt{-\ell})$ can be expressed as $h(-\ell) =…

A curious identity of Bunyakovsky (1882), made more widely known by P\'olya and Szeg{\H o} in their ``Problems and Theorems in Analysis", gives an evaluation of a sum of the floor function of square roots involving primes $p\equiv…

Let $p_{1}, p_{2}$ be two distinct prime integers, let $n$ be a positive integer, $n$$\geq 3$ and let $\xi_{n} $ be a primitive root of order $n$ of the unity. In this paper we obtain a complete characterization for a quaternion algebra…

Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest number of consecutive quadratic non-residues modulo p is always less…

In this paper, we show that if $p\equiv 1\pmod 4$ is prime, then $4F_p$ admits a representation of the form $u^2-pv^2$ for some integers $u$ and $v$, where $F_n$ is the $n$th Fibonacci number. We prove a similar result when $p\equiv -1\pmod…

A. Booker and C. Pomerance (2017) have shown that any residue class modulo a prime $p\ge 11$ can be represented by a positive $p$-smooth square-free integer $s = p^{O(\log p)}$ with all prime factors up to $p$ and conjectured that in fact…

Let $p$ be a prime divisor of the order of a finite group $G$. Then $G$ has at least $2 \sqrt{p-1}$ complex irreducible characters of degrees prime to $p$. In case $p$ is a prime with $\sqrt{p-1}$ an integer this bound is sharp for…

In this paper, we construct infinitely many quadruples of real quadratic fields whose class numbers are all divisible by $3$. To the best of our knowledge, this is the first result towards the divisibility of the class numbers of certain…

It is shown that the quartic Fermat equation $x^4 +y^4=1$ has nontrivial integral solutions in the Hilbert class field $\Sigma$ of any quadratic field $K=\mathbb{Q}(\sqrt{-d})$ whose discriminant satisfies $-d \equiv 1$ (mod 8). A corollary…

In this note we revisit classic work of Soundararajan on class groups of imaginary quadratic fields. Let $A,B,g \ge 3$ be positive integers such that $\gcd(A,B)$ is square-free. We refine Soundararajan's result to show that if $4 \nmid g$…

This article deals with a conjecture, introduced in [GQ] (hereinafter $SFLT2$), which generalizes the second case of Fermat's Last Theorem: {\it Let $p>3$ be a prime. The diophantine equation $\frac{u^p+v^p}{u+v}=w_1^p$ with $u,v,u+v,…

In this article, we consider primes $p \equiv 5 \pmod 8$ and are able to prove that $p \equiv 5 \pmod {16}$ if $2p$ is a congruent number.

Let $Q(x,y)$ be a quadratic form with discriminant $D\neq 0$. We obtain non trivial upper bound estimates for the number of solutions of the congruence $Q(x,y)\equiv\lambda \pmod{p}$, where $p$ is a prime and $x,y$ lie in certain intervals…

Let $\ell\geq 5$ be prime. Let $\mathcal{F}_\ell$ be the collection of (isomorphism classes of) pure number fields $\mathbb{Q}(\sqrt[\ell]{a})$ of degree $\ell$, ordered by the absolute value of their discriminant. In 2018, Benli proved a…

For a prime $p\equiv 1 \,(\bmod{4})$, let \[ \varepsilon = \frac{1}{2}\left( t + u\sqrt{p}\right) \] be the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{p})$. In 1951, N. Ankeny, E. Artin, and S. Chowla asked whether $p$…