相关论文: Cube sums of form $3p$ and $3p^2$ II

Let $p\equiv 2,5\mod 9$ be an odd prime. In this paper, we prove that at least one of $3p$ and $3p^2$ is a cube sum by constructing certain nontrivial Heegner points. We also establish the explicit Gross-Zagier formulae for these Heegner…

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 nonzero rational number is called a cube sum if it is of form $a^3+b^3$ with $a,b\in \mathbb{Q}^\times$. In this paper, we prove that for any odd integer $k\geq 1$, there exist infinitely many cube-free odd integers $n$ with exactly $k$…

We use Zagier's one-sentence proof approach to show that a prime number $p$ admits a form $p=a^2+ab+b^2$ for some integers $a$ and $b$ if and only if $p=3$ or $p\equiv 1 \pmod{3}$.

In this note, we use integral binary cubic forms to study the rational cube sum problem. We prove (unconditionally) that for any positive integer $d$, infinitely many primes in each of the residue classes $ 1 \pmod {9d}$ as well as $ -1…

Let H(N) denote the Hurwitz class number. It is known that if $p$ is a prime, then {equation*} \sum_{|r|<2\sqrt p}H(4p-r^2) = 2p. {equation*} In this paper, we investigate the behavior of this sum with the additional condition $r\equiv…

Let $p\equiv 8\mod 9$ be a prime. In this paper we give a sufficient condition such that at least one of $p$ and $p^2$ is the sum of two rational cubes. This is the first general result on the $8$ case of the so-called Sylvester conjecture.

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 $p$ be a prime greater than $3$ and let $a$ be a rational p-adic integer. In this paper we try to determine $\sum_{k=1}^{[p/3]}\binom{3k}ka^k\pmod p$, and real the connection between cubic congruences and the sum…

We show that almost all primes $p\not\equiv \pm 4 \bmod{9}$ are sums of three cubes, assuming a conjecture due to Hooley, Manin, et al. on cubic fourfolds. This conjecture is approachable under standard statistical hypotheses on geometric…

Let $p \geq 5$ be a prime and for $a, b \in \mathbb{F}_{p}$, let $E_{a,b}$ denote the elliptic curve over $\mathbb{F}_{p}$ with equation $y^2=x^3+a\,x + b$. As usual define the trace of Frobenius $a_{p,\,a,\,b}$ by \begin{equation*}…

We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum…

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,…

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 examine what integers are representable as sums of three cubes. We also provide formulas for the number of representations of $x^3+y^3+z^3=n$ under the condition $x+y+z=t$. Also we show how the problem of three cubes is related to…

We produce congruences modulo a prime $p>3$ for sums $\sum_k\binom{3k}{k}x^k$ over ranges $0\le k<q$ and $0\le k<q/3$, where $q$ is a power of $p$. Here $x$ equals either $c^2/(1-c)^3$, or $4s^2/\bigl(27(s^2-1)\bigr)$, where $c$ and $s$ are…

Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper, based on the work of Brillhart and Morton, by using the work of Ishii and Deuring's theorem for elliptic curves with complex multiplication we solve some…

It is well known that when a prime $p$ is congruent to 1 modulo 4, the sum of the quadratic residues equals the sum of the quadratic nonresidues. In this note we give analogous results for the case where $p$ is congruent to 3 modulo 4.

We consider certain families of integers $n$ determined by some congruence condition, such that the global root number of the elliptic curve $E_{-432n^2}: Y^2=X^3-432n^2$ is $1$ for every $n$, however a given $n$ may or may not be a sum of…

In this paper, we prove that every prime $p$ which is congruent to $4,7$ modulo $9$ is the sum of two rational cubes. This is $2/3$ of Sylvester's conjecture which has a history of nearly 150 years since 1879. In the proof, we use recent…