Rational -parallelogram envelopes via -congruent elliptic curves.

We introduce a new generalization of -congruent numbers by defining a rational -parallelogram envelope. It turns out that the existence of a rational -parallelogram envelope associated to an n is connected to the property of n being a -congruent number. Then, we study some problems related to the rational -parallelogram envelopes, using the arithmetic of algebraic curves. Our results generalize the recent work of Ochiai, where only the case = π / 2 was considered. Moreover, we answer open questions raised by Ochiai in the more general setting of an arbitrary Pythagorean angle. [ABSTRACT FROM AUTHOR]