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13. Note on Pythagorean Triangles
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47 13 Note on Pythagorean Triangles c. 1890 Houghton Library A Pythagorean triangle is a set of 3 integer numbers proportional to the legs and hypotheneuse of a right triangle. It is irreducible if the 3 integers have no common measure. The number of irreducible Pythagorean triangles of which a given number is hypotheneuse is 0, if the number contains a prime factor not of the form (4n 1); otherwise, it is equal to 2p 1 where p is the number of its different prime factors. For example, 725 52 29. Accordingly, It is also true that (500) 2 (525) 2 (725) 2 ; but that triangle is reducible . Again, 1105 5 13 17. Accordingly, (333) 2 110889 (364) 2 132496 (644) 2 414736 (627) 2 393129 (725) 2 525625 (725) 2 525625 (47) 2 2209 (264) 2 69696 (1104) 2 1218816 (1073) 2 1151329 (1105) 2 1221025 (1105) 2 1221025 (576) 2 331776 (744) 2 553536 (943) 2 889249 (817) 2 667489 (1105) 2 1221025 (1105) 2 1221025 ...