(put
start value of 'n' or subsequent integer value)

x =
(only pick up integer values)

Some Interesting Properties of
Pythagorean sub-triplets

1. y is
always less than n/2 i.e. y<n/2

2.n is always greater than 2. lowest integer value of n cannot be
less than 3.

3.For a given y, the smallest value of n of the triplet
cannot be less than 2y+1.

4.For a given y, several Pythagorean sub-pairs (n, x) can be found out.

5.The lowest (n,x) pair are such that 3x=3y=n (exception
y=8)

6.For y=odd no. , n/y
3,5,7,9,11,... which is an AP with common difference 2.
(exception y=9 where 1st,4th,7th,10th...no. with common difference of 3 are
in ratio 5/2,7/2.9/2 for n/y)

7.For y=even no.
, n/y 3,4,5,6,7,... which is an AP with common difference 1.
(exception y=8 where 2nd,4th,6th,8th...no.
are in above ratio for n/y
and
& 1st,3rd,5th,7th.. ..no. are with n/y
5/2,7/2,9/2,11/2...with common difference of 1)

8.
If y=odd number, [y n x]_{ y=} y [y n x]_{ 1 }i.e. y
times the corresponding value of sub-triplets for 1
except y=9 or odd multiple of 9.For odd multiples of 9, [y n x]_{ 9m=}
m [y n x]_{ 9 (where m=y/9 & is odd.)}

9. If y is a multiple of 8, [y n x]_{ y}=( y/8) [y n
x]_{ 8 }i.e.( y/8)times the corresponding value of sub-triplets for
8.

10. For given y, there is a constant difference
between
consecutive n values. For y=1,2 diff=2; y=4,8 diff=4;
y=3,6,9 diff=6; y=5,10 diff=10 and y=7 diff=14.
All differences are of even value.

11.The sub-triplets of "y=an odd integer" are all odd. The sub-triplets of
"y=even integer" all even with exception of
y=8
and its odd multiples where x is an odd integer.

12. For each integer value of y, infinite number of triplets can
be generated.

13.If a (equals n), b (equals n plus x) are the smaller sides and c
(equals n plus x plus y) is the hypotenuse of the right angled triangle in
X-Y plane, then a2+b2=c2.If c (equals n') be the smaller side of
another right angled triangle and d (equals n' plus x') is a line along
z-axis, then the consequent triangle for the same y value exists
and (x',n') can be found provided n'=x+n+y and x' satisfies the equation
n'=y'+(2y^{2}+2yx)^{1/2 } and n'=y'+(2y^{2}+2yx)^{1/2
}and x' is an integer or a rational number.Thus a2+b2+d2=f2 where f=n'+x'+y'.

Find Pythagorean sub-triplets (y, n, x) with given y
and given pair number (up to y=20)

(pair no. is 1 for the starting pair for a given y.)

Next Level of triplets (explore with above largest number as the smaller
side of another right angled triangle in 3-D space)

n'=
smallest number:

y'=
middle number:

x'=
largest number:

(y, n, x chart at different y values)

Pair ----------->

1

2

3

4

5

6

7

8

9

10

Y=1 --->n

Y=1 --->x

Y=2 ----->n

Y=2 ----->x

Y=3 ----->n

Y=3 ----->x

Y=4 ----->n

Y=4 ----->x

Y=5 ----->n

Y=5 ----->x

Y=6 ----->n

Y=6 ----->x

Y=7 ----->n

Y=7 ----->x

Y=8 ----->n

Y=8 ----->x

Y=9 ----->n

Y=9 ----->x

Y=10 ---->n

Y=10 ---->x

Y=11 ---->n

Y=11 --->x

Y=12 ---->n

Y=12 --->x

Y=13 ---->n

Y=13 --->x

Y=14 ---->n

Y=14 ---->x

Y=15 ---->n

Y=15 --->x

Y=16 ---->n

Y=16 --->x

Y=17 ---->n

Y=17 --->x

Y=18 ---->n

Y=18 --->x

Y=19 ---->n

Y=19 --->x

Y=20 ---->n

Y=20 --->x

Y=21 ---->n

Y=21 --->x

Y=22 ---->n

Y=22 --->x

Y=23 ---->n

Y=23 --->x

Y=24 ---->n

Y=24 --->x

Y=25 ---->n

Y=25 --->x

Y=26 ---->n

Y=26 --->x

Y=27 ---->n

Y=27 --->x

Y=28 ---->n

Y=28 --->x

Y=29 ---->n

Y=29 --->x

Y=30 ---->n

Y=30 --->x

Y=31 ---->n

Y=31--->x

Y=32 ---->n

Y=32 --->x

Y=33 ---->n

Y=33 --->x

Y=34 ---->n

Y=34 --->x

Y=35 ---->n

Y=35 --->x

Y=36 ---->n

Y=36 --->x

in a circle, the diameter subtends a
right angle at any point on the circumference, and the connected
lengths of 3 sides of the right angled triangle constitute Pythagorean
Triplets (not necessarily integers) and For every pair of (n',x'), there exists (n",x") such
that n"=x' and x"=n'. y value is maximum for n equal to
total/2^1/2.The deductions for determining the lengths are
here.Now taking one end of the diameter as
the origin(0,0) of the cartesian co-ordinate system, the co-ordinates
(p,q) of any point on
circumference of the circle are hereby called principal Pythagorean
co-ordinates or principals which are infinite in number for a given
value of n+x+y. Angle of the
point is reckoned from origin.

Analogous Pythagoras Theorem for Hyperbolic
right angled triangle

If a, b and c are the three sides of a
hyperbolic
triangle (right angled) with angle between two small sides a, b being 90° and c
is the hypotenuse, then coshc=cosha * coshb; It
may be noted that the vertices of the triangle lie in the hyperbolic plane.