In number theory, Pythagorean Triples PT and congruent numbers are closely connected. The latter are natural numbers A which can be deduced by rational PTs a.b.ceQ! whose area is Aq2=ab/2 with A, qEN. They are known since ancient times and a matter of research also in our days. The formulae for their construction are known at least since Euclid, but the decision whether a given number is a congruent number and especially its representation are not so simple.

In 11 chapters, after an introduction, Pythagorean Triples and congruent numbers, the representation of natural numbers, elementary characteristics of congruent numbers, other characteristics of congruent numbers, special congruent numbers, complex congruent numbers, Fermat's Last Theorem, congruent numbers and elliptic curves, characteristics of elliptic curves, and theorems and conjectures related to congruent numbers are discussed.