PPT (3,4,5) and its multiples

 
A Pythagorean triple consists of three positive integers   a,   b,   and   c,   such that
a^2 \; + \; b^2 \; = \; c^2
 

Let’s take the Primitive Pythagorean triple   (3, 4, 5)

Note that,       5^4 = 4^4 + 4^4 + 3^4 + 2^4 + 2^4

or,     2\times 2^4 \; + \; 3^4 \; + \; 2\times 4^4 \; = \; 5^4

The multiples of   (3, 4, 5) :     n\times (3, \; 4, \; 5)   satisfy the equation

2 \, k^4 \; + \; a^4 \; + \; 2 \, b^4 \; = \; c^4

c^4 \; = \; (a^2 + b^2)^2 \; = \; a^4 \; + \; b^4 \; + \; 2 \, a^2 \, b^2
2 \, k^4 \; + \; a^4 \; + \; 2 \, b^4 \; = \; a^4 \; + \; b^4 \; + \; 2 \, a^2 \, b^2
k^4 \; = \; (a \, b)^2 \; - \; b^4/2
With   (3, 4, 5)
k^4 \; = \; (3\times 4)^2 \; - \; 4^4/2 \; = \; 2^4

with the multiples of   (3, 4, 5):

(a \, b \, n^2)^2 \; - \; (b \, n)^4/2
= \; a^2 \, b^2 \, n^4 \; - \; (b^4 \, n^4)/2
= \; n^4 \,(a^2 \, b^2 \; - \; (b^4)/2)

 

4th POWERS

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Advertisements

About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s