You are here:

Advanced Math/Pythagorean triples

Advertisement

Expert: - 2/2/2011


Question

Question 2 c
Dear Paul,

Can you help, please?
PS see the attached JEPG document.

Thanks
Aba

Answer
First part is routine:

x^2 + y^2 = 25:

(a/b)^2 + (c/d)^2 = 25

(ad)^2 + (cd)^2 = 25(bd)^2  << mult by b^2 d^2

(ad)^2 + (cd)^2 = (5bd)^2  << mult by b^2 d^2

.................................................

For any triple, there are integers  s,t  such that

s^2 - t^2,  2st,  s^2 + t^2  are a PT

SEE:  http://en.wikipedia.org/wiki/Pythagorean_triple

Assign:    ad = s^2 - t^2
and let:   bc = 2st
and then:  5bd = s^2 + t^2



ad = s^2(1 - (t/s)^2)  << factoring  s^2
bc = s^2(2t/s)         << factoring  s^2
5bd = s^2(1 + (t/s)^2) << factoring  s^2

Ready to go:

bc    s^2(2t/s)
--- = -----------------   << divide
5bd    s^2(1 + (t/s)^2)


c      (2t/s)
--- = ----------------   << reduce
5d     (1 + (t/s)^2)


c     5(2t/s)
--- = ---------------- = y
d     (1 + (t/s)^2)

.....................

ad    s^2(1 - (t/s)^2)
--- = -----------------  << divide
5bd    s^2(1 + (t/s)^2)

a    (1 - (t/s)^2)
--- = ---------------  << reduce
5b    (1 + (t/s)^2)

a    5(1 - (t/s)^2)
--- = --------------- = x
b    (1 + (t/s)^2)


Finito.

Paul Klarreich

Expertise

I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

Experience

I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.

Education/Credentials
-----------

©2012 About.com, a part of The New York Times Company. All rights reserved.