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The Pythagorean Suits


Number theory is the area of ​​mathematics that investigates deep and subtle relationships between positive integers. Pythagoras and his followers linked these numbers to geometry, and thus began one of the most successful strands of number theory, namely the binomial: arithmetic and geometry. Around 1700 BC tables were found in Babylon containing lists of whole number suits with the property that one of the numbers squared equal to the sum of the squares of the other two. Because such lists were extensive, it is believed that the Babylonians already had a systematic method of generating such suits. There are historical records that prove the existence and use of such tables in ancient Egypt. Consider the squares of the natural numbers 1², 2², 3², 4², 5²,… If we take the sum of two squares, we will eventually get another square. The most famous example of this fact is: 3² + 4² = 5², but there are other examples: 5² + 12² = 13², 20² + 21² = 29², and many others. However 2² + 3² = 13 is not a square. Therefore, it is natural to ask if there is an infinite number of Pythagorean suits. The answer is yes and the reason is very simple: if (x, y, z) is a Pythagorean tender, then by multiplying it by a positive integer c, we get (cx, cy, cz) which is a new Pythagorean tender, therefore, (cx) ² + (cy) ² = c² (x² + y²) = c²z² = (cz) ². On the other hand these suits are not the most interesting and so we define primitive suits, ie those where a, b, and c have no common factor and satisfy the relation x² + y² = z².

On the other hand, the Pythagoreans were interested in the right triangles whose collars have integer length x and y and the length z of the hypotenuse relates to x and y so that z² = x² + y². Such a relationship is the famous Pythagorean Theorem. The search for all positive integers that satisfy the identity x² + y² = z² is equivalent to the problem of determining all right triangles whose sides are integers.

The Pythagoreans were, around 600 BC, the first to give a method of determining infinite suits of this kind, today called the Pythagorean suits. Using current notation we describe the method as follows: let x = n, y = 1 (n² -1), z = 1 (n² + 1) where n is an odd integer greater than 1; so the resulting tendon (x, y, z) is a Pythagorean terna where z = y + 1. Note some examples: 3² + 4² = 5², 5² + 12² = 13², 7² + 24² = 25², 9² + 40² = 41², 11² + 60² = 61². Note that there are other suits than these: for example, when z = y + 2, ie 8² + 15² = 17², 12² + 35² = 37², 16² + 63² = 99², 20² + 65² = 101². The philosopher Plato (430 - 349 BC) has found another method for determining all these tendencies, which in modern notation are the formulas: x = 4n², y = 4n² -1, z = 4n² +1. The Greek mathematician Tales of Miletus brought about a substantial change in knowledge when he turned mathematics that had been practiced as some form of numerology into a deductive science. Around 300 BC, when Euclid published the collection of 13 books called Elements, all the mathematical facts presented were formally demonstrated. In the tenth book Euclid gave a method of obtaining all Pythagorean suits. Although not presenting a formal demonstration of his method, Euclid obtained all the suits. Using current notation, the method consists of the following formulas: x = t (a²-b²), y = 2tab, z = t (a² + b²) where t, a, and b are arbitrary positive integers such that a> b, a and b have no factors in common, and if a is odd then b is even and vice versa. This completely solves the natural problem of knowing which ones are all the Pythagorean suits.

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