The complete solution of a diophantine equation involving biquadrates

AbstractWhile parametric solutions of the diophantine equation ∑i=1sxi4=∑i=1syi4 are known for any integral value of s⩾2, the complete solution in integers is not known for any value of s. In this paper, we obtain the complete solution of this equation when s⩾13

Extraction of nth roots of 2×2 matrices

AbstractThis paper is concerned with the determination of algebraic formulae giving all the solutions of the matrix equation Xn=A where n is a positive integer greater than 2 and A is a 2×2 matrix with real or complex elements. If A is a 2×2 scalar matrix, the equation Xn=A has infinitely many solutions and we obtain explicit formulae giving all the solutions. If A is a non-scalar 2×2 matrix, the equation Xn=A has a finite number of solutions and we give a formula expressing all solutions in terms of A and the roots of a suitably defined nth degree polynomial in a single variable. This leads to very simple formulae for all the solutions when A is either a singular matrix or a non-singular matrix with two coincident eigenvalues. Similarly when n=3 or 4, we get explicit algebraic formulae for all the solutions. We also determine the precise number of solutions in various cases