相关论文: Proof of a conjecture of Guy on class numbers

For a prime $p\equiv 3$ $(\text{mod }4)$, let $h(-8p)$ and $h(8p)$ be the class numbers of $\mathbb{Q}(\sqrt{-2p})$ and $\mathbb{Q}(\sqrt{2p})$, respectively. Let $\Psi(\xi)$ be the Hirzebruch sum of a quadratic irrational $\xi$. We show…

For any distinct two primes $p_1\equiv p_2\equiv 3$ $(\text{mod }4)$, let $h(-p_1)$, $h(-p_2)$ and $h(p_1p_2)$ be the class numbers of the quadratic fields $\mathbb{Q}(\sqrt{-p_1})$, $\mathbb{Q}(\sqrt{-p_2})$ and…

For fixed $q\in\{3,7,11,19, 43,67,163\}$, we consider the density of primes $p$ congruent to $1$ modulo $4$ such that the class group of the number field $\mathbb{Q}(\sqrt{-qp})$ has order divisible by $16$. We show that this density is…

Let $p$ be a prime. The $2$-primary part of the class group of the pure quartic field $\mathbb{Q}(\sqrt[4]{p})$ has been determined by Parry and Lemmermeyer when $p \not\equiv \pm 1\bmod 16$. In this paper, we improve the known results in…

The density of primes $p$ such that the class number $h$ of $\mathbb{Q}(\sqrt{-p})$ is divisible by $2^k$ is conjectured to be $2^{-k}$ for all positive integers $k$. The conjecture is true for $1\leq k\leq 3$ but still open for $k\geq 4$.…

Let $p$ be a prime and let $S_2(\Gamma(p))$ be the space of weight $2$ cusp forms for the principal congruence subgroup $\Gamma(p)$. Then $\mathrm{SL}_2(\mathbb{F}_p)$ acts on $S_2(\Gamma(p))$ in a natural way. Around 1928, Hecke proved…

Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2\pi/p) and C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2\pi/p) are proved. For example, T(p) equals p times…

Let $h_{(m,k)}$ be the class number of $\mathbb{Q}(\sqrt{1-2m^k}).$ We prove that for any odd natural number $k,$ there exists $m_0$ such that $k \mid h_{(m,k)}$ for all odd $m > m_0.$ We also prove that for any odd $m \geq 3,$ $k \mid…

Given a random real quadratic field from $\{ \mathbb{Q}(\sqrt{p}\,) ~|~ p \text{ primes} \}$, the conjectural probability $\mathbb{P}(h=q)$ that it has class number $q$ is given for all positive odd integers $q$. Some related conjectures of…

We use a variant of Vinogradov's method to show that the density of the set of prime numbers $p\equiv -1\bmod~4$ for which the class group of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-8p})$ has an element of order $16$ is…

A formula for the class number $h$ of the imaginary quadratic field $Q(\sqrt{-p}$ is obtained by counting on a specific way the quadratic residues of a prime number of the form $p=4n-1.$ Formulas for the sum of the quadratic residues are…

We determine the $3$-class groups of $\mathbb{Q}(\sqrt[3]{p})$ and $K=\mathbb{Q}(\sqrt[3]{p},\sqrt{-3})$ when $p\equiv 4,7\bmod 9$ is a prime and $3$ is a cubic modulo $p$. This confirms a conjecture made by Barrucand-Cohn, and proves the…

In this paper, we consider sums of class numbers of the type $\sum_{m\equiv a\pmod{p}} H(4n-m^2)$, where $p$ is an odd prime, $n\in \mathbb{N},$ and $a\in \mathbb{Z}$. By showing that these are coefficients of mixed mock modular forms, we…

We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime $p\equiv 1\pmod 4$ and integer $a\not\equiv0\pmod p$, we prove that \begin{align*}&(-1)^{|\{1\le k<\frac p4:\ (\frac kp)=-1\}|}\prod_{1\le…

Let $p$ be an odd prime. In this paper we investigate quadratic residues modulo $p$ and related permutations, congruences and identities. If $a_1<\ldots<a_{(p-1)/2}$ are all the quadratic residues modulo $p$ among $1,\ldots,p-1$, then the…

For a prime $p>3$, let $D$ be the discriminant of an imaginary quadratic order with $|D|< \frac{4}{\sqrt{3}}\sqrt{p}$. We research the solutions of the class polynomial $H_D(X)$ mod $p$ in $\mathbb{F}_p$ if $D$ is not a quadratic residue in…

The main purpose of this paper is to find all the prime numbers p for which whenever we add to p an odd square less than p we obtain a number which has at most two different prime factors. We solve completely the cases $p\equiv 1,3,5 \pmod…

If $p$ is an odd prime, then we prove that $\e(H_2(G,\mathbb{Z})) \mid p\ \e(G)$ for $p$ groups of class 7. We prove the same for $p$ groups of class at most $p+1$ with $\e(Z(G))=p$. We also prove Schurs conjecture if $\e(G/Z(G))$ is $2,3$…

A computational proof is given for congruences modulo $p$ for the class equation $H_{-28p}(X)$, when the prime $p$ satisfies $p \equiv 3$ (mod $4$), and for the product $H_{-7p}(X) H_{-28p}(X)$, when $p \equiv 1$ (mod $4$).

Let $P_{r}$ denote an integer with at most $r$ prime factors counted with multiplicity. In this paper we prove that for some $\lambda < \frac{1}{12}$, the inequality $\{\sqrt{p}\}<p^{-\lambda}$ has infinitely many solutions in primes $p$…