An integer $n$ is congruent if there is a triangle with rational sides whose area is $n$. In the 1980s Tunnell gave an algorithm to test congruence which relied on counting integral points on the ellipsoids $2x^2+y^2+8z^2 = n$ and $2x^2+y^2+32z^2=n$. The correctness of this algorithm is conditional on the conjecture of Birch and Swinnerton-Dyer. The known methods for testing Tunnell's criterion use $O(n)$ operations. In this paper we give several methods with smaller exponents, including a randomized algorithm using time $n^{1/2 + o(1)}$ and space $n^{o(1)}$, and a deterministic algorithm using space and time $n^{1/2 + o(1)}$.

We use the computer algebra system Magma to study graded rings of Fano 3-folds of index $\ge 3$ in terms of their Hilbert series.

In this paper, we consider the Tate and Ate pairings for the genus-$2$ supersingular hyperelliptic curves $y^{2}=x^{5} -\alpha x$ ($\alpha = \pm2$) defined over finite fields of characteristic five. More precisely, we construct a distortion map explicitly, and show that the map indeed gives an input for which the value of the Tate pairing is not trivial. We next describe a computation of the Tate pairing by using the proposed distortion map. We also see that this type of curve is equipped with a simple quintuple operation on the Jacobian group, which leads to an improvement for computing the Tate pairing. We further show the Ate pairing, a variant of the Tate pairing for elliptic curves, can be applied to this curve. The Ate pairing yields an algorithm which is about $50\,\%$ more efficient than the Tate pairing in this case.

A positive integer $n$ is said to be \textit{harmonic} if the harmonic mean $H(n)$ of its positive divisors is an integer. Ore proved that every perfect number is harmonic and conjectured that there exist no odd harmonic numbers greater than $1$. In this article, we give the list of all harmonic numbers up to $10^{14}$. In particular, we find that there exist no nontrivial odd harmonic numbers less than $10^{14}$.

This paper introduces a fast algorithm for solving the DLP of Jacobian of hyperelliptic curve of small genus. To solve the DLP, Gaudry first shows that the idea of index calculus is effective, if a subset of the points of the hyperelliptic curve of the base field is taken by the smooth elements of index calculus. In an index calculus theory, a special element (in our case it is the point of hyperelliptic curve), which is not a smooth element, is called a large prime. A divisor, written by the sum of several smooth elements and one large prime, is called an almost smooth divisor. By the use of the almost smooth divisor, Thériault improved this index calculus. In this paper, a divisor, written by the sum of several smooth elements and two large primes, is called a $2$-almost smooth divisor. By use of the $2$-almost smooth divisor, we are able to give more improvements. The algorithm of this attack consists of the following seven parts: 1) Preparing, 2) Collecting reduced divisors, 3) Making sufficiently large sets of almost smooth divisors, 4) Making sufficiently large sets of smooth divisors, 5) Solving the linear algebra, 6) Finding a relation of collected reduced divisors, and 7) Computing a discreet logarithm. Parts 3) and 4) need complicated eliminations of the large prime, which is the key idea presented within this paper. Before the tasks in these parts are completed, two sub-algorithms for the eliminations of the large prime have been prepared. To explain how this process works, we prove the probability that this algorithm does not work to be negligible, and we present the expected complexity and the expected storage of the attack.

Let $A$ be a principally polarized Abelian surface defined over $\mathbb{Q}$ with $\End(A)=\mathbb{Z}$ and $\widetilde{A}$ be the reduction at a good prime $p$. In this paper, we study the density of prime numbers $p$ for which $\widetilde{A}(\mathbb{F}_p)$ is a cyclic group and establish a conjecture which relates this density.