On elliptic curves y=x−nx with rank zero

In this paper we determine all elliptic curves E_n:y²=x³−n²x with the smallest 2-Selmer groups S_n=Sel₂(E_n(Q))={1} and S_n′=Sel₂(E_n′(Q))={±1,±n}(E_n′:y²=x³+4n²x) based on the 2-descent method. The values of n for such curves E_n are described in terms of graph-theory language. It is well known that the rank of the group E_n(Q) for such curves E_n is zero, the order of its Tate-Shafarevich group is odd, and such integers n are non-congruent numbers.     

On the Diophantine Equation x+q=y

In this paper it has been proved that if q is an odd prime, q?7 (mod 8), n is an odd integer ?5, n is not a multiple of 3 and (h,n)=1, where h is the class number of the filed Q(√−q), then the diophantine equation x²+q^2k+1=yⁿ has exactly two families of solutions (q,n,k,x,y).     

Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves

This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevich-Tate group. Unable to compute that, we computed the five other quantities and solved for the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2-torsion of the Shafarevich-Tate group, which we could compute.

     

On the diophantine equation x(x + 1)(x + 2)…(x + (m − 1)) =g(y)

Let g(y) ? Q[Y] be an irreducible polynomial of degree n ≥ 3. We prove that there are only finitely many rational numbers x, y with bounded denominator and an integer m ≥ 3 satisfying the equation x(x + 1) (x + 2)…(x + (m − 1) ) = g(y). We also obtain certain finiteness results when g(y) is not an irreducible polynomial.     

On the number of divisors of x2 + x + A

Let A be a positive or negative rational integer such that integers in the field of √1 ? 4A have unique prime factorization. An elementary criterion will be obtained for x² + x + A to be a prime number, where x is a positive integer. The criterion implies that for positive A the polynomial x² + x + A is prime for x = 0, 1,…, A ? 2.     

Special values of L-functions of elliptic curves over Q and their base change to real quadratic fields

For an elliptic curve E over Q, and a real quadratic extension F of Q, satisfying suitable hypotheses, we study the algebraic part of certain twisted L-values for E/F. The Birch and Swinnerton-Dyer conjecture predicts that these L-values are squares of rational numbers. We show that this question is related to the ratio of Petersson inner products of a quaternionic form on a definite quaternion algebra over Q and its base change to F.     

On Mordell's Equation

In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals Q. Here we illustrate our method by applying it to Mordell's Equation y²=x³⁺k for 0 k Z and draw some conclusions from our numerical findings. In fact we solve Mordell's Equation in Z for all integers k within the range 0 < | k | 10 000 and partially extend the computations to 0 < | k | 100 000. For these values of k, the constant in Hall's conjecture turns out to be C=5. Some other interesting observations are made concerning large integer points, large generators of the Mordell–Weil group and large Tate–Shafarevi groups. Three graphs illustrate the distribution of integer points in dependence on the parameter k. One interesting feature is the occurrence of lines in the graphs.     

Regions of absolute stability of explicit Runge-Kutta-Nyström methods for y″ = f(x, y, y′)

We examine regions of absolute stability of s-stage explicit Runge-Kutta-Nyström (R-K-N) methods of order s for s = 2, 3, 4 for y″ = f(x, y, y′) by applying these methods to the test equation: y″ + 2αy′ + β²y = O, α, β ? 0, α + β > 0. We also examine the regions of absolute stability of Runge-Kutta (R-K) methods for first order differential equations of respective orders. Interestingly, it turns out that regions of absolute stability of R-K methods and R-K-N methods of respective orders for which the asymptotic relative error does not deteriorate are identical. Our present investigations are in continuation of the recent results of Chawla and Sharma [1].     

On the number of coprime integral solutions of y2 = x3 + k

Let N′(k) denote the number of coprime integral solutions x, y of y² = x³ + k. It is shown that lim sup_k→∞N′(k) ≥ 12.     

关于椭圆曲线E_D:y~2=x~3-D~2x的BSD猜想

本文证明了形如ｙ~２＝ｘ~３－ｐ-１~２…ｐ-ｎ~２ｘ的一类椭圆曲线的Ⅲ群２-分支在一定条件下同构于 Ｚ／２Ｚ × Ｚ／２Ｚ,其阶为 ４,与Ｌ函数部分的相应结果和 Ｒｕｂｉｎ Ｋ．关于有复乘的椭圆曲线的重要结果一起,我们知道ＢＳＤ猜想对本文定理中的椭圆曲线成立．     

On the Birch-Swinnerton-Dyer Conjecture of Elliptic Curves E D : y 2 = x 3−D 2 x

Abstract We prove in this paper that the BSD conjecture holds for a certain kind of elliptic curves. Project supported by the National Natural Science Foundation of P. R. China     

On the period function of x″+f(x)x′+g(x)=0

We study the period function T of a center O of the title's equation. A sufficient condition for the monotonicity of T, or for the isochronicity of O, is given. Such a condition is also necessary, when f and g are odd and analytic. In this case a characterization of isochronous centers is given. Some classes of plane systems equivalent to such equation are considered, including some Kukles’ systems.     

On the g-circulant solutions to the matrix equation Am = λJ, II

Let g and n be positive integers and let $\text{k =}\text{n(g, n)}\text{(g}{}^{\mathrm{m}}\text{, n)}$. If θ(x) is a multiple of Σ_{i = 0}^{k ? 1}xⁱ, then the g-circulant whose Hall polynomial is equal to θ(x) satisfies the matrix equation in the title. If the g-circulant whose Hall polynomial is equal to Σ_{i = 0}^{h ? 1}xⁱ satisfies the matrix equation in the title, then h is a multiple of k.     

关于方程xy+yz+zx=n的正整数解

本文在广义Ｒｉｅｍａｎｎ猜想成立的条件下证明了：当且仅当正整数ｎ＝１，２，４，６，１０，１８，２２，３０，４２，５８，７０，７８，１０２，１３０，１９０，２１０，３３０，４６２时，方程ｘｙ＋ｙｚ＋ｚｘ＝ｎ无正整数解（ｘ，ｙ，ｚ）．