Pythagorean triangles

Among the infinite number of possible rectangular triangles, the so-called "Pythagorean triangles", whose sides are integers, always attracted special interest. Undoubtedly, "Pythagorean triangles" belong to the category of "treasures of geometry", and their search is one of the most interesting pages in the history of mathematics. The most widely known of these is a rectangular triangle with sides 3, 4 and 5. It was also called "sacred" or "Egyptian", as it was widely used in Egyptian culture.

Among the infinite number of possible rectangular triangles, the so-called Pythagorean triangles

Since the equation x2 + y2 = z2 is homogeneous, when multiplying x , y and z, other Pythagorean triangles are obtained for the same natural number. The Pythagorean triple (x, y, z) is called primitive if it can not be obtained in this way from some other Pythagorean triple, then is, x, y, and z are mutually prime numbers. In other words, the greatest common divisor (x, y, z) is equal to the number one. The more primitive Pythagorean triples, the more Pythagorean triangles with their lengths approaching an isosceles triangle. It follows that an infinitely large primitive Pythagorean triple is the side of an infinitely large isosceles triangle.

For the "Egyptian" triangle, the Pythagorean theorem takes the following numerical form: 4² + 3² = 5². After the Pythagorean theorem was discovered, the question arose how to find all the "Pythagorean triangles" - triples of natural numbers that can be sides of a right triangle. Some common methods of searching for such triples of numbers, for example those mentioned above (3, 4, 5) or (5, 12, 13), were known to the Babylonians. One of the cuneiform tablets contains "Pythagorean triangles" consisting of 15 triples. Among them there are consisting of so large numbers that there can be no question of finding them by selection.