JournalslemVol. 57, No. 1/2pp. 91–102

Quadratic Form made a Perfect Power by a New Composition Theorem on Arbitrary Quadratic Forms

  • Ajay Choudhry

    Old J.N.U. Campus, New Delhi, India
Quadratic Form made a Perfect Power by a New Composition Theorem on Arbitrary Quadratic Forms cover
Download PDF

Abstract

This paper deals with the diophantine equation Q(x1,x2,,xm)=ynQ(x_1,\,x_2,\,\ldots,\,x_m)= y^n, where mm and nn are arbitrary positive integers and Q(x1,x2,,xm)Q(x_1,\,x_2,\,\ldots,\,x_m) is an arbitrary quadratic form in the mm variables x1,x2,,xmx_1,\,x_2,\,\ldots,\,x_m. While solutions of special cases of this equation have been published earlier, the general equation of this type has not been solved till now. To solve this equation, we first show that, given an arbitrary quadratic form Q(x1,x2,,xm)Q(x_1,\,x_2,\,\ldots,\,x_m) in mm variables, there exists a {\it composition formula} Q(ui)Q2(vi)=Q(wi)Q(u_i)\,Q^2(v_i)=Q(w_i) where uiu_i and viv_i (i=1,2,,mi= 1,\,2,\,\ldots,\,m) are arbitrary variables and the wiw_i (i=1,2,,mi=1,\,2,\,\ldots,\,m) are cubic forms in the variables uiu_i and viv_i (i=1,2,,mi=1,\,2,\,\ldots,\,m). This is a new composition formula, different from the standard composition formulae of the type Q(ui)Q(vi)=Q(wi)Q(u_i)Q(v_i)=Q(w_i) which are known for certain classes of quadratic forms. As the equation Q(xi)=ynQ(x_i)=y^n is not always solvable, we prove a theorem giving a necessary and sufficient condition for its solvability. We use the aforementioned composition formula to obtain parametric solutions of the equation Q(xi)=ynQ(x_i)=y^n, and also give some numerical examples.

Cite this article

Ajay Choudhry, Quadratic Form made a Perfect Power by a New Composition Theorem on Arbitrary Quadratic Forms. Enseign. Math. 57 (2011), no. 1/2, pp. 91–102

DOI 10.4171/LEM/57-1-4