Fibonacci's Method
Leonardo of Pisa (c. 1170 – c. 1250) described this method for generating primitive triples using the sequence of consecutive odd integers, and the fact that the sum of the first terms of this sequence is . If is the -th member of this sequence then .
Choose any odd square number from this sequence ( ) and let this square be the -th term of the sequence. Also, let be the sum of the previous terms, and let be the sum all terms . Then we have established that and we have generated the primitive triple . This method produces an infinite number of primitive triples, but not all of them.
EXAMPLE: Choose . This odd square number is the fifth term of the sequence, because . The sum of the previous 4 terms is and the sum of all terms is giving us and the primitive triple = .
Read more about this topic: Formulas For Generating Pythagorean Triples
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