Find the minimum odd integer(positive) such that:
(1) for some positive integers .
(2) for some positive integers .
(3) .
From (2), is a Pythagorean triple. If this triple is primitive then there are integers s, t such that , and . So we can try taking and . In that case, (3) says that . Therefore . This says that is a square triangular number. The smallest such number apart from 1 is . That suggests taking t = 6 and s – t = 9, so that s = 15.
That gives the solution , with . This certainly satisfies all the conditions (1), (2), (3), but I do not know whether it is the smallest solution apart from a = 5.