D(n)-quintuples with square elements

Elements with Square Roots in Finite Groups

In this paper, we study the probability that a randomly chosen element in a finite group has a square root, in particular the simple groups of Lie type of rank 1, the sporadic finite simple groups and the alternating groups.

On group Elements Having Square Roots

A Note on Diophantine Quintuples

Introduction. Diophantus noted that the rational numbers 1/16, 33/16, 17/4 and 105/16 have the following property: the product of any two of them increased by 1 is a square of a rational number (see [2, 3]). Let n be an integer. A set of positive integers {a1, a2, . . . , am} is said to have the property D(n) if aiaj + n is a perfect square for all 1 ≤ i < j ≤ m. Such a set is called a Diophant...

The Number of Diophantine Quintuples

A set {a1, . . . , am} of m distinct positive integers is called a Diophantine m-tuple if aiaj + 1 is a perfect square for all i, j with 1 ≤ i < j ≤ m. It is known that there does not exist a Diophantine sextuple and that there exist only finitely many Diophantine quintuples. In this paper, we first show that for a fixed Diophantine triple {a, b, c} with a < b < c, the number of Diophantine qui...

Hopf Bifurcation with Dn-symmetry

The aim of this paper is to study Hopf bifurcation with Dn-symmetry assuming Birkhoff normal form. We consider the standard action of Dn on C. This representation is absolutely irreducible and so the corresponding Hopf bifurcation occurs on C ⊕ C. Golubitsky and Stewart (Hopf bifurcation with dihedral group symmetry: Coupled nonlinear oscillators. In: Multiparameter Bifurcation Series (M. Golub...