A Property of Primitive Pythagorean triples

JOARDAR, BODHIDEEP (2017) A Property of Primitive Pythagorean triples. At Right Angles, 6 (2). pp. 67-68.

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Abstract

A primitive Pythagorean triple, or PPT for short, is a triple ( a , b , c ) of coprime positive integers satisfying the relation a 2 + b 2 = c 2 . Some well-known PPTs are: ( 3 , 4 , 5 ) , ( 5 , 12 , 13 ) and ( 8 , 15 , 17 ) . See Box 1 for some basic facts about PPTs. A note on PPTs There have been several articles in past issues of At Right Angles exploring PPTs and ways of generating them. Here are some features about PPTs which you need in this article (we invite you to provide proofs): If (a, b, c) is a PPT, then: (i) c is odd; (ii) one out of a, b is odd and the other one is even; (iii) the even number in {a, b} is a multiple of 4 . We agree to list the numbers in the PPT so that odd number and b is the even number. a is the The following property is worth noting: b is a multiple of 4 . To see why, write b 2 = c 2 −a 2 . Note that a and c are odd, and recall that any odd square is of the form 1 (mod 8) . This implies that b 2 is a multiple of 8 and hence that b is a multiple of 4 . (If b were even but not a multiple of 4 , then b 2 would be a multiple of 4 but not a multiple of 8 .) This article focuses on one particular family of PPTs, those having b = c − 1 . For this family we have: so: a 2 + (c − 1) 2 = c 2 , c = a 2 + 1 , 2 ∴ a 2 = 2c − 1, b = a 2 − 1 . 2

Item Type: Articles in APF Magazines
Uncontrolled Keywords: Pythagorean triple, primitive, divisibility, modulus, proof
Subjects: Natural Sciences
Natural Sciences > Mathematics
Divisions: Azim Premji University > University publications > At Right Angles
Depositing User: Mr. Sachin Tirlapur
Date Deposited: 15 Sep 2018 10:52
Last Modified: 15 Sep 2018 10:52
URI: http://publications.azimpremjifoundation.org/id/eprint/1349
Publisher URL: http://apfstatic.s3.ap-south-1.amazonaws.com/s3fs-...

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