In the 1940s, Otto E Neugebauer and Abraham J Sachs, mathematics historians, pointed out that the other three columns were essentially

Pythagorean triples " sets of integers, or whole numbers, that satisfy the equation a2 + b2 = c2.

The tree of primitive

Pythagorean triples is no different (Figure 1).

The author covers

Pythagorean triples, the Monty Hall problem and other probability theory puzzles, the Lucas sequence, the triangular numbers, the square numbers, and many other related subjects.

Keywords: Trigonometry,

Pythagorean Triples, Vedic mathematics.

This notwithstanding, it is still very difficult to predict

Pythagorean triples like (a, b, c) or (x, y, r) in (4).

Appendices show representations of Pythagoras and his theorem in art, list many proofs and provide a table of

Pythagorean triples.

2], which means S(a), S(b, z are

Pythagorean triples, and GCD(S(a), S(b)) = d > 1, z | [d.

In [5] the conditions on x, y and z for

Pythagorean triples (x, y, z) where x, y, z are considered in small modes are determined.

Pythagorean triples are sets of three whole numbers, a, b, and c, that can be the lengths of the sides of a right triangle, meaning that they fit the formula [a.

They have variously interpreted the cryptic columns of numbers, written in the wedge-shaped script called cuneiform, as a trigonometric table or a sophisticated scheme for generating

Pythagorean triples.

Using the Phantom Square, we will now show that all

Pythagorean triples can be built from multiples of the (3,4,5) triangle.

Today we call them primitive

Pythagorean triples where the term primitive implies that the side lengths share no common divisor.