Integral Points on the Circle X2 + Y2 = c

Abstract

Let us begin with the equation, $${X^2} + {Y^2} = 1$$ (1) which represents the unit circle, over any number field. Regarding this equation, Dipendra Prasad asked the following questions. What are all the number fields K JOT which, equation (1) has a non trivial integral solution? Suppose that K is a number field for which, equation (1) has a nontrivial integral solution. Can we determine all such solutions? Note that (±1,0) and (0, ±1) are trivial integral solutions of (1). If C denotes the set of all (x, y) satisfying (1), then C is an abelian group under the composition, $$\left( {{x_1},{y_1}} \right)\left( {{x_2},{y_2}} \right): = \left( {{x_1}{x_2} - {y_1}{y_2},{x_1}{y_2} + {y_1}{x_2}} \right).$$ In [PS1], we proved the following theorem, which determines the structure of this group in terms of the number of complex imbeddings of K, and thus provides a complete answer to the above questions.