Pythagorean triples have been first mentioned in the second millennium BC by Babylonians. The triple consists of three positive integers a, b and c, which are related to each other by the Pythagorean theorem a2 + b2 = c2. Thus, they represent the length of the three sides of a right Pythagorean triangle. Primitive Pythagorean triples are the triples, which are co-prime, therefore the only common positive divisor is 1.
One of the methods for generating Pythagorean triples was proposed by Leonardo of Pisa. His technique uses consequent odd integers, such as 1 and 3, then add them together in order to generate the third one. Similar procedure is used for the 4th value. In the next step, 2nd and 3rd numbers are multiplied and their product is then doubled to obtain the 1st member of the Pythagorean triple. The product f the 1st and the 4th numbers gives the 2nd member of the triple. The sum of squares of the 3rd and the 4th values produces last value (Knott, 2011).
1) If 1 and 3 are selected, 1 + 3 = 4, 3 + 4 = 7, which generates the sequence 1,3,4,7.
Thus, 3 ×4 = 12 , 2 × 12 = 24 – first value, 1× 7 = 7 – the second value, 32 + 42 = 25 – third value. Therefore the Pythagorean triple is (24, 7, 25).
Verification through Pythagorean theorem: 242 + 72 = 576 + 49 = 625 = 252
2) If 3 and 5 are selected, 5 + 3 = 8, 8 + 5 = 13, which generates the sequence 3,5,8, 13.
Thus, 5 ×8 = 40 , 2 × 40 = 80 – first value, 3 × 13 = 39 – the second value, 52 + 82 = 89 – third value. Therefore the Pythagorean triple is (80, 39, 89).
Verification through Pythagorean theorem: 802 + 392 = 6400 + 1521 = 7921 = 892
3) If 5 and 7 are selected, 5 + 7 = 12, 7 + 12 = 19, which generates the sequence 5, 7, 12, 19.
Thus, 7 × 12 = 84 , 2 × 84 = 168 – first value, 5 × 19 = 95 – the second value, 72 + 122 = 193 – third value. Therefore the Pythagorean triple is (168, 95, 193).
Verification through Pythagorean theorem: 952 + 1682 = 9025 + 28224 = 37249 = 1932
4) If 7 and 9 are selected, 7 + 9 = 16, 9 + 16 = 25, which generates the sequence 7, 9, 16, 25.
Thus, 9 × 16 = 144 , 2 × 144 = 288 – first value, 7 × 25 = 175 – the second value, 92 + 162 = 337 – third value. Therefore the Pythagorean triple is (288, 175, 337).
Verification through Pythagorean theorem: 2882 + 1752 = 82944 + 30625= 113569= 3372
5) If 9 and 11 are selected, 9 + 11 = 20, 11 + 20 = 31, which generates the sequence 9, 11, 20, 31. Thus, 11 × 20 = 220 , 2 × 220 = 440 – first value, 9 × 31 = 279 – the second value, 112 + 202 = 521 – third value. Therefore the Pythagorean triple is (440, 279, 521).
Verification through Pythagorean theorem: 4402 + 2792 = 193600+ 77841= 271441= 5212
The problem described above deals with the generation of Pythagorean triplets, the numbers, which are related to each other through the Pythagorean Theorem. The method chosen for generating uses Fibonacci rule and converts 2 consecutive odd integers into a triple through several algebraic expression. The choice of this method was dictated by the ease of its implementation and its comprehensiveness. The knowledge of Pythagorean triples allows to simplify geometric calculations involving right triangles providing a shortcut to determining the sides of right triangles, which have the lengths equal to a positive integers.
References
Knott, R. (2011, July 05). Pythagorean triples and fibonacci numbers. Retrieved from
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html
