 Subject: More on Pythagorean triples, II
Date: Sun, 5 Apr 1998 15:59:14 -0300
From: Gerald Briggs

Were you aware that there is a special irrational number which is linked to a set of pythagorean triples which I, for lack of any other nomenclature I have found on this, have named the perfect-primitive-pythagorean triples. The first triangle that belongs to this set is (3,4,5) and the second is (20,21,29). The third such triangle is (119,120,169) and the fourth is (696,697,985). These are pythagorean triples (a,b,c) by the fact that a^2 + b^2 = c^2, and a,b, and c are integers. The distinct feature here is that each triangle's two non-hypoteneuse sides are exactly one apart. If you take the number (2^0.5 + 1)^2 and multiply it onto the hypoteneuse of the last triple (ie 5.8284268 * 985 = 5741.0003) you find with near perfect precision the next larger hypoteneuse corresponding with the next larger perfect-primitive-pythagorean triple (4059,4060,5741). The 4059 and 4060 are found by dividing the 5741 by the square root of 2 and adding a half to get 4060.0002 and subtracting a half to get 4059.0002. The next precision is even better (5.8284268 * 5741= 33460.999) yielding 33461 as the next hypoteneuse. In essence, what seems peculiar is that irrational numbers are being used to approximate integers instead of vice-versa. What kind of field of mathematics can prove that these triangles exist in such perfect order? Is there any depth to this phenomenon? 