Pythagorean Triples

Most people know the "3-4-5 triangle" of the Pythagoras trio, and each draft kit has a right triangle (and 45-45-90). This triangle is different from most right-angled triangle because it has three integer edges. Pythagorean theorem tells us that the sum of the squares of the sides of a right triangle gives the square of the hypotenuse: 32 + 42 = 52 Whether this relationship is unique or another right angle with 3 integers I often ask if there is a triangle. The same can be said about the sides.

Pythagorean theorem can describe several special forms. The triad of Pythagoras is a right triangle whose side length and oblique side are integers. The smallest Pythagoras triple is a triangle with a = 3, b = 4, and c = 5. Using Pythagorean theorem, you can see 9 + 16 = 25. Theorem squares can also be literal; if you use each length of a right triangle as the edge of a rectangle, the edge rectangle will be the same area as the square created by the length of the hypotenuse.

The history of this theorem is also related to the discovery of Pythagorean triad. Pythagoras triad is made up of three positive integers like $ c ^ 2 = a ^ 2 + b ^ 2 $. Since $ 5 ^ 2 = 3 ^ 2 + 4 ^ 2 $, the three integers 3, 4, and 5 are known as the Pythagorean triad. Pythagoras trio was discovered by Babylonians between 2000 BC and 1786 BC.

The Pythagorean equation x 2 + y 2 = z 2 has infinite number of positive integer solutions for x, y, and z, these solutions are known as Pythagoras trio. By around 1637, Fermat wrote on the edge of the book that if n is an integer greater than 2, the more general equation a + bn = cn does not have a positive integer solution. He insisted that he had general evidence of his speculation, but Fermat could not find his evidence he did not leave details of his evidence. His argument was discovered in 30 years after his death. This assertion is later called Fermat's last theorem and has not been solved in mathematics for the next three and a half and a half.

Well, you are definitely thinking. Let's see the real thing. Our first simple matter is whether there are an infinite number of Pythagoras triads, ie triples that satisfy the natural numbers (a, b, c) of equation a 2 + b 2 = c 2. The answer is "yes". This is a very stupid reason. Take the Pythagorean triad (a, b, c) and multiply it by another number d to get a new Pythagoras triple (da, db, dc). This is true as it is not difficult to prove that these guesses are correct. First, if both a and b are even, c is an even number. This means that triple is not original because the common factor of a, b, c is 2. Next assume that both a and b are odd. This means that c must be an even number. This means that there are such numbers x, y, z.