The non-extensibility of D(4k)-triples {1, 4k(k-1), 4k^2+1} with |k| prime

For a nonzero integer n, a set of m distinct positive integers {a1, ... , am} is called a D(n)-m-tuple if aiaj+n is a perfect square for each i, j with 1 ≤ i < j ≤ m. Let k be an integer with |k| prime. Then we show that the D(4k)-triple {1, 4k(k-1), 4k^2+1} cannot be extended to a D(4k)-quadruple

The D(-k2)-triple {1,k2+1,k2+4} with k prime

Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a prime number. In this paper we prove that the D(-k2)-triple {1,k2+1,k2+4} cannot be extended to a D(-k2)-quadruple if k≠3. And for k=3 we prove that if the set {1,10,13,d} is a D(-9)-quadruple, then d=45

Computation of Equilibria in OLGModels with Many Heterogeneous Households

This paper develops a decomposition algorithm by which a market economy with many households may be solved through the computation of equilibria for a sequence of representative agent economies. The paper examines local and global convergence properties of the sequential recalibration (SR) algorithm. SR is then demonstrated to efficiently solve Auerbach- Kotlikoff OLG models with a large number of heterogeneous households. We approximate equilibria in OLG models by solving a sequence of related Ramsey optimal growth problems. This approach can provide improvements in both efficiency and robustness as compared with simultaneous solution methods.Computable general equilibrium, Overlapping generations, Microsimulation, Sequential recalibration

Walking Alone: My Career in Mathematics

In this article, dictated by Maohua Le and arranged by Yongzhong Hu, Professor Le briefly recounts his legendary experience of self-study mathematics, which reflects the life experiences of his generation of Chinese people

On the 2-part of class groups and Diophantine equations

This thesis contains several pieces of work related to the 2-part of class groups and Diophantine equations. We first give an overview of some techniques known in computing the 2-part of the class groups of quadratic number fields, including the use of the Rédei symbol and Rédei reciprocity in the study of the 8-rank of the class groups of quadratic fields. We review the construction of governing fields for the 8-rank by Corsman and extend a proof of Smith on the distribution of the 8-rank for imaginary quadratic fields, to real quadratic fields, conditional on the general Riemann hypothesis. In joint work with Peter Koymans, Djordjo Milovic, and Carlo Pagano, we improve a previous lower bound by Fouvry and Klüners, on the density of the solvability of the negative Pell equation over the set of squarefree positive integers with no prime factors congruent to 3 mod 4. We show how Rédei reciprocity allows us to apply techniques introduced by Smith to obtain this improvement. In joint work with Djordjo Milovic, using Kuroda's formula, we study the average behaviour of the unit group index in certain families of totally real biquadratic fields Q(√p,√d) parametrised by the prime p. In joint work with Christine McMeekin and Djordjo Milovic, we study certain cyclic totally real number fields K, in which we attach a quadratic symbol spin(a,σ) to each odd prime ideal a and each non-trivial σ in Gal(K/Q). We prove a formula for the density of primes ideals p such that spin(p,σ) = 1 for all non-trivial σ in Gal(K/Q). Finally, we study integral points on the quadratic twists E_D:y²=x³-D²x of the congruent number curve. We show that the number of non-torsion integral points on E_D is << (3.8)^{\rank E_D(Q)} and its average is bounded above by 2. We deduce that the system of simultaneous Pell equations aX²-bY²=d, bY²-cZ²=d for pairwise coprime positive integers a,b,c,d, has at most << (3.6)^{ω(abcd)} integer solutions

Complete solutions of the simultaneous Pell's equations (a2+2)x2−y2=2 (a^2+2)x^2-y^2 = 2 and x2−bz2=1 x^2-bz^2 = 1

In this paper, we consider the simultaneous Pell equations $(a^2+2)x^2-y^2 = 2$ and $x^2-bz^2 = 1$ where $a$ is a positive integer and b > 1 is squarefree and has at most three prime divisors. We obtain the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most one solution in positive integers. When a solution exists, assuming the positive solutions of the Pell equation $(a^2+2)x^2-y^2 = 2$ are $x = x_m$ and $y = y_m$ with $m\geq 1$ odd, then the only solution of the system is given by $m = 3$ or $m = 5$ or $m = 7$ or $m = 9$

Restricted 132-Dumont permutations

A permutation $\pi$ is said to be {\em Dumont permutations of the first kind} if each even integer in $\pi$ must be followed by a smaller integer, and each odd integer is either followed by a larger integer or is the last element of $\pi$ (see, for example, \cite{Z}). In \cite{D} Dumont showed that certain classes of permutations on $n$ letters are counted by the Genocchi numbers. In particular, Dumont showed that the $(n+1)$st Genocchi number is the number of Dummont permutations of the first kind on $2n$ letters. In this paper we study the number of Dumont permutations of the first kind on $n$ letters avoiding the pattern 132 and avoiding (or containing exactly once) an arbitrary pattern on $k$ letters. In several interesting cases the generating function depends only on $k$.Comment: 12 page