HenrykDot

★ HenrykDot.com ★
is the online companion to a series of books published by AIUT under the common main title
"Physics of My Imaginary Space-Time" by Henryk Dot.

POLSKI

last update
20/01/2023

Home

Physics 3 - Maxwell
Note from the Author
Table of Contents
What is this book
Historical facts
New aspects
Fully erroneous
Incorrectly interpreted

Physics 3 - Chapter 1
Equations
Complex vectors form
The Most General form
The General Solution

Physics 3 - Chapter 2
Solutions
Initial conditions
Non-homogeneous equation
Solution for three-directions
The four laws

Physics 3 - Supplement
Fermat's proof
Beal's conjecture
Pythagorean triples
Inertial mass
Gravity constans big G
What does the Moon look at?

Physics 3 - Final notes
Final notes

Physics 4 - New book
Entry

Books published by AIUT
are found in libraries according to the list of compulsory copies.

Second Edition of "Fizyka 3"
ISBN 978-83-926856-1-6
Fizyka

can be bought in Warsaw
in the Academic Bookstore
PW Publishing House
Noakowskiego street 18/20

and in Katowice
in the bookstore "Liber"
Bankowa street 11.
(area of Silesian University)

English edition of "Physics"
ISBN 978-83-926856-2-3
Fizyka

is also in libraries
and the distribution method should be asked wydawca@aiut.com.

3.1.2. Pythagorean triples Returning to our presented proof – as a “side effect” it allows also for easy way to find possible solutions of (3.1) for N = 2 (a.k.a. Pythagorean triples). New variable x along with auxiliary variable e = c - a which were introduced in our proof lead to the following relations:
b²=((c - a)+x)²=(e+x)²=e²+2‧ e‧ v+x² and	(3.9)
b²=c²- a²=(e+a)²- a²=e²+2‧ e‧ v	(3.10)
Comparing (3.9) and (3.10) we obtain:
x²=2‧ (a - x) ,	(3.11)
where it can be seen that x must be an even number. We can introduce new integral variable y such that x=2‧ y, then inserting it into (3.11) we obtain:
.	(3.12)
We can conclude then e and y have to be expressed by following relations: e=2‧ r_e² and y=r_e‧ r_y , where r_e is a divisor of e and r_y is a second divisor of y, besides r_e. If this were not a case, a, e and y would have common divisor, then this would mean that a and b were not relatively prime. In this case we obtain:
,	(3.13)
where r_y cannot be an even number, since 2 is a divisor of e. It should be then assumed that: r_y=2‧ r_j+1 , where r_j - is an integer ⩾ 0, which results in: e=2‧ r_e² , x=2‧ r_e‧ (2‧ r_j+1) ,
a=r_y‧ (r_y+2‧ r_e)= (2‧ r_y+1)‧ (2‧ (r_j+r_e)+1) ,	(3.15a)
b=2‧ r_e‧ (2‧ r_j+1)+2‧ r_e²)= 2‧ r_e‧ (2‧ r_j+r_e+1) ,	(3.15b)
c=r_y‧ (r_y+2‧ r_e)+2‧ r_e²= (2‧ r_j+1)‧ (2‧ (r_j+r_e)+1)+2‧ r_e²	(3.15c)
So, we can re-write them as: a=(2‧ r_y+1)‧ 2‧ (r_j+r_e)+1)= (2‧ r_j+r_e+1) ² - r_e² , b=2‧ r_e‧ (2‧ r_j+1)+2‧ r_e²)= 2‧ (2‧ r_j+r_e+1)‧ r_e , c=(2‧ r_j+1)‧ (2‧ (r_j+r_e)+1)+2‧ r_e²= (2‧ r_j+r_e+1)²+r_e² and then they correspond to well-known publicized relations
a=m²- n² , b=2‧ m‧ n , c=m²+n² .	(3.16)
Our derived relations (3.15) have additional advantage since there is no restrictions such as in case of relation (3.16) where m and n cannot be odd numbers at the same time.