Pythagorean Triples Essay

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Pythagorean Triples Jessica Scallions 1-30-11 MAT 126 Tracy Abram Pythagorean Triples In order to understand Pythagorean Triples, we must first understand what a Pythagorean Theorem is. The Pythagorean Theorem states that for any right triangle, the sum of the squares of the length of the legs of a right triangle is equal to the square of the length of the hypotenuse (the side opposite the right triangle) (Bluman, 2005 p.471). If a, b, and c are positive integers, they are together called Pythagorean Triples. The smallest such Pythagorean Triple is 3, 4 and 5. It can be seen that 32 + 42 = 52 (9+16=25). I am going to investigate Pythagorean Triples where the shortest side is an odd number and all 3 sides are positive integers. I will then investigate other families of Pythagorean Triples to see if Pythagoras' theorem (a²+b²=c²) works. A Pythagorean triple consists of three positive integers, a, b, and c, such that a2 + b2= c2. A well-known example is 3, 4, and 5 because 32+ 42 = 9 + 16 = 25 =52. Other examples include (5, 12, 13); (8, 15, 17); and (20, 48, 52). It is easy to see that there are an infinite number of Pythagorean triples because one can always multiply all three positive integers by a positive integer k and get another Pythagorean Triple. It can be observed that a Pythagorean Triple always consists of: all even numbers, or two odd numbers and an even number. A Pythagorean Triple can never be made up of all odd numbers or two even numbers and one odd number. This is true because: the square of an odd number is an odd number and the square of an even number is an even number and the sum of two even numbers is an even number and the sum of an odd number and an even number is in odd number. Therefore, if one of a and b is odd and the other is even, c would have to be odd. Similarly, if both a and b are even, c would be an even

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