Squares in certain recurrent sequences and some Diophantine equations

Abstract

The object of this thesis is to solve, in integers X and Y, various equations of the form [equation] where d and N are given square free integers. The work stems from two papers by J.H.E. Cohn in which the equations [equation] are solved for certain values of d. It is well known that the solutions of X2-dY2=4, and those of X2-dY2 = -4 where such solutions exist, may be expressed in terms of the least positive solutions of these equations. Solutions of the equations [equations] may now be sought among those of x2-dY2 = +1,+4. Extensive work in this direction has been done by W. Ljunggren working in the quadratic field R(d2) and other allied algebraic fields. His methods are powerful but deep and complicated. It is possible to show that the solutions of the equations X2 for sequences which satisfy a three-term recurrence relation. By applying the elementary theory of quadratic residues to these equations Cohn has solved the equations and for those d for which either of the equations has solutions (X,Y) for which X and Y are both odd. This thesis extends Cohn's work, Using similar methods, to solve the equations and for the same values of d as above and any given integer N. A few limited results are given for other values of d. L.J. Mordell has given simple conditions under which the equation can have no solutions. A theorem of a similar type concerning the equations x4-dY2 = 1,4 is proved. Finally the results proved in this thesis are compared with those of Cohn, Ljunggren and Mordell.