Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Pythagorean triples
Replies: 8   Last Post: Mar 21, 2013 2:53 AM

 Messages: [ Previous | Next ]
 Doctor Nisith Bairagi Posts: 27 From: Uttarpara, West Bengal, India Registered: 3/2/13
Re: Pythagorean triples
Posted: Mar 13, 2013 3:45 AM

> There are some Pythagorean triples ( = integers
> {a,b,c}: a^2+b^2-c^2=0) such that the shorter two
> sides differ by only 1. E.g. 20, 21, 29 ; 119, 120,
> 169.
>
> Is there a finite number of such triples? If so, how
> many?
>
> Or
>
> Show that there is an infinite number.

From: Doctor Nisith Bairagi.
Date : March 13, 2013
Subject: SPECIAL PYTHAGOREAN TRIPLES: GENERAL FORMULAE
Re: Pythagorean Triples(posted : June 26, 2006)

Dear Mathematicians,
I have read the interesting question posed by Jason Osborne (07/10/97) and subsequent discussion by Doctor Rob (07/11/97) 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 a straight forward 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:

(1) Proposed Sequence:
u- sequence : 1, 2, 5,12, 29, 70, 169, 408, 985, 2378, ?(for n = 1,2,3?.).
[start with u(1) = 1, [u(n) = 2u(n-1) + u(n-2)], (169 = 2 x 70 + 29)]

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:

v(n) = [((2)^(0.5) +1)^n)/(2(2)^(0.5))
+ (-1)^(n-1).((2)^(0.5) -1)^n)/(2(2)^(0.5))]
u(n) = v(n-1)

z(n) = [(5(2)^(0.5)-7).((3-2(2)^(0.5))^(n-1)) / (2(2)^(0.5))
+ (-1)^(n-1).(5(2)^(0.5)+7).((-3-2(2)^(0.5))^(n-1)) / (2(2)^(0.5))].

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 (1) the proposed sequences, or, by (2) 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].

From:
Doctor Nisith Kumar Bairagi
My email: <bairagi605@yahoo.co.in>

??????????????????????????????..

Date Subject Author
6/26/06 cuthbert
6/26/06 Philippe 92
6/26/06 cuthbert
6/26/06 Philippe 92
6/26/06 cuthbert
3/13/13 Doctor Nisith Bairagi
3/14/13 Doctor Nisith Bairagi
3/21/13 shyamal kumar das
3/21/13 shyamal kumar das