From: Doctor Nisith Kumar Bairagi. My email address : firstname.lastname@example.org Date : March 13, 2013 Subject: SPECIAL PYTHAGOREAN TRIPLES: GENERAL FORMULAE
Dear Mathematicians, Dear Philippe92 and Cuthbert, I have read the interesting question posed by Jason Osborne (07/10/97), Philippe92 and Cuthbert, and subsequent discussion by Doctor Rob (07/11/97, and also discussion between you two) on the special type of Pythagorean triples of the type (119,120,169), (696,697,985), ?, in which the two adjacent sides of a right triangle forming right angles, are consecutive numbers, (i.e., x and y differ by 1), while z is the hypotenuse, satisfying x^2 + y^2 = z^2.
I send another straightforward and valid answer to this quarry (omitting the derivation part) as follows: By putting: x = (u^2 ? v^2) = (u + v)(u- v), y = 2uv, and z = (u^2 + v^2), and putting numerical values for u and v, the infinite sets of all Pythagorean triples (including the special type in question), can be written down directly.
Here, we present this special category of Pythagorean triples in this form: (u,v : x, y, z) through (1) the sequences of (u, v, and z), and also through (2) the proposed formulae, as follows:
v- sequence : 2, 5,12, 29, 70, 169, 408, 985, 2378, 5741, ?(for n = 1,2,3?.). [start with v(1) = 2, from the same u-sequence, [(v(n) = u((n+1)), (v(8) = u(9) = 2378)].
The x-sequence and y-sequence can be easily calculated from the u and v-sequence. It will be noted that all the numbers of the x-sequence are odd and factorable. Again, if the n-th term is even, x >y, we get x - y =1, and if odd, y >x, we get y ? x =1. This is controlled by : /x ? y/ = (-1)^n, for any even or odd value of n.
z- sequence : 5, 29, 169, 985, 5741, 33461, 195025, ?(for n = 1,2,3?.). [start with z (1) = 5, and dropping alternate terms in u-sequence, [z(n) = 6z(n-1)- z(n-2), (5741 = 6x 985 - 169)], Also: [(z(n) = v(2n) = u(2n+1)), (z(5) = v(10) = u(11) = 5741)].
(2) Proposed Formula: Without resorting to the sequence form, the terms u(n), v(n), and z(n) can be directly calculated as:
Thus, for n = 9, the 9-th triple is: (u, v: x, y, z) = (2378, 985, 4684659, 4684660, 6625109).
For larger and larger terms, the ratio B = z(n+1)/z(n) converges to [3+2(2)^0.5], or, its reciprocal 1/B = z(n)/z(n+1)to [3-2(2)^0.5]. This ratio number B (or its reciprocal 1/B), enjoys the property that: (B + 1/B) = 6. Compare this property with that of Fibonacci sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, ?,where the difference of F = F(n+1)/F(n) = [(1 + (5)^0.5] / 2, and its reciprocal 1/F = F(n)/F(n+1)= [(-1 + (5)^0.5] / 2, yields: (F -1/F) = 1.
All the infinite sets of special Pythagorean triples, in which out of the three, the two are consecutive numbers, can be obtained easily either by the proposed sequences, or, by the direct application of the formulae, as shown here.
[For further details, refer to the Book ?Advanced Trigonometric Relations through Nbic Functions? by Nisith K Bairagi, New Age International Publishers, New delhi (2012), Appendix A] Does this solution satisfy the required enquiries of Mr Jason Osborne and Doctor Rob? Please acknowledge/accept my proposed solution, and produce this solution in your column for other readers, and reply your comments to me in my Email.
From: Doctor Nisith Kumar Bairagi My email: <email@example.com>