The Pythagorean Theorem - Educational Outreach

There are three Pythagorean-triple types: primitive (P),primitive twin (PT), and composite (C). The definitions foreach are as follows.1. Primitive: A Pythagorean Triple ( a , b,c)where there is nocommon factor for all three positive integers a , b,&c .2. Primitive twin: A Pythagorean Triple ( a , b,c)where thelongest leg differs from the hypotenuse by one.3. Composite: A Pythagorean Triple ( a , b,c)where there is acommon factor for all three positive integers a , b,&c .The definition for primitive twin can help us find anassociated m & n condition for identifying the same. Ifa 2mn is the longest leg, thenc a 1mm22 n2 2mn n( m n)2 2mn12 1 1 m n 1Examining Table 3.2 confirms the last equality; primitivetwins occur whenever m n 1. As a quick exercise, weinvite the reader to confirm that the two cases m, n 8, 7and m , n 9, 8 also produce Pythagorean twins withc 100 .b m2 n2Ifis the longest leg [such as the case form 7 & n 2 ], thenc b 1m22n n221 m2 n21..93