Related papers: Proof of a conjecture of Guy on class numbers

Let $p$ be a prime number such that $p=2$ or $p\equiv 1\pmod 4$. Let $\varepsilon_p$ denote the fundamental unit of $\mathbb{Q}(\sqrt{p})$ and let $a$ be a positive square-free integer. The main aim of this paper is to determine explicitly…

Let $p\equiv3\pmod{4}$ be a prime and $r$ a positive integer. We show that $$ \prod_{k=1}^{(p^{2r}-1)/2}\frac{4k-1}{4k+1}\equiv1\pmod{p^2}. $$ This confirms a recent conjecture of Guo.

We give a simple matrix-based proof of congruence equations modulo a prime $p$ involving sums of binomial coefficients appearing in Pascal's triangle. These equations can be used to construct some groups of exponent $p^n$. These groups, as…

Let $a\geq 1$ and $n>1$ be odd integers. For a given prime $p$, we prove under certain conditions that the class groups of imaginary quadratic fields $\mathbb{Q}(\sqrt{a^2-4p^n})$ have a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$. We…

Let $p$ be an odd prime and let $d\in\{2,3,7\}$. When $(\frac{-d}p)=1$ we can write $p=x^2+dy^2$ with $x,y\in\mathbb Z$; in this paper we aim at determining $x$ or $y$ modulo $p^2$. For example, when $p=x^2+3y^2$, we show that if $p\equiv…

In the present paper we obtain new upper bound estimates for the number of solutions of the congruence $$ x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in\cU, $$ for certain ranges of $H$ and $|\cU|$, where $\cU$ is a…

For an odd prime $p$ and polynomial $P(T)$, we consider the extension $F$ of $k={\mathbb F}_p(T)$ defined by adjoining a root of $x^p+Tx-P(T)$. Such a field is a function field analogue of the number field ${\mathbb Q}(\sqrt[p]{n})$. We…

On d\'emontre une conjecture due \`a N. Kuhn concernant la cohomologie singuli\`ere \`a coefficients mod p des espaces, comme module instable sur l'alg\`ebre de Steenrod. Notre d\'emonstration de ce r\'esultat, d\'ej\`a connu en…

Suppose that $ m\equiv 1\mod 4 $ is a prime and that $ n\equiv 3\mod 4 $ is a primitive root modulo $ m $. In this paper we obtain a relation between the class number of the imaginary quadratic field $ \Q(\sqrt{-nm}) $ and the digits of the…

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) =…

Much is known about binomial coefficients where primes are concerned, but considerably less is known regarding prime powers and composites. This paper provides two conjectures in these directions, one about counting binomial coefficients…

A conjecture of Mordell states that if $p$ is a prime and $p$ is congruent to $3$ mod $4$, then $p$ does not divide $y$ where $(x,y)$ is the fundamental solution to $x^{2}-py^{2}=1$. The conjecture has been verified for primes not exceeding…

In this paper, we confirm some congruences conjectured by V.J.W. Guo and M.J. Schlosser recently. For example, we show that for primes $p>3$, $$…

We consider sums of Hurwitz class numbers of the type $\sum_{t \equiv m \pmod{M}}H(4p-t^2)$, where $M$ is composite. For $M=6$ and $8$, we show that these sums can be expressed in terms of coefficients of CM cusp forms. This leads to…

We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\sum_g(\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\ldots,p-1$,…

Let $p\equiv 4,7\mod 9$ be a rational prime number such that $3\mod p$ is not a cubic residue. In this paper we prove the 3-part of the product of the full BSD conjectures for $E_p$ and $E_{3p^3}$ is true using an explicit Gross-Zagier…

A formula for the sum of quadratic residues modulus a prime $p=4n-1$ is studied. We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new concepts based on these roots. A number of formulas for…

Let $p$ be an odd prime and let $d$ be an integer not divisible by $p$. We prove that $$ \prod_{1\le m,n\le p-1\atop p\nmid m^2-dn^2}\ (x-(m+n\sqrt{d})) \equiv \begin{cases}\sum_{k=1}^{p-2}\frac{k(k+1)}2x^{(k-1)(p-1)}\pmod p &\text{if}\…

We prove a $p$-converse theorem for elliptic curves $E/\mathbb{Q}$ with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$ in which $p$ is ramified. Namely, letting $r_p =…

Assume $x,\ y,\ n$ are positive integers and $n$ is odd. In this note, we show that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{x^{2}-y^{n}})$ is divisible by $n$ for fixed $x, n$ if $\gcd(2x,y)=1$ and $y>C$ where…