Pythagorean triplets
Pythagoras was a Greek  mathematician and a philosopher & a mysterious figure about whose life very little is known except that he first propounded the now famous Pythagorean theorem and founded Pythagorean School of philosophy which firmly advocated that the cosmos could be described in terms of whole numbers. When subsequently they hit upon irrational numbers, the discovery was kept under wraps for fear of contradiction to their philosophy till one of their followers leaked it out. The story goes that the poor fellow was thrown to the high seas and died of drowning. Interestingly Pythagorean theorem was expressed in terms of geometry and not in terms of numbers. We also know that the theorem does not hold good for Non-Euclidian space. Pythagoras traveled extensively and covered Egypt, Arabia, Babylon and possibly India and was influenced by the philosophical thoughts of these countries. He died in 475 B.C. at the age of 75.

      For today's featured topic!    Click here

sum of the square of two smaller sides of a right angled triangle equals square of the third side i.e. hypotenuse.

sum of the square of two rational numbers equals square of the third number. These numbers are called Pythagorean triplets.

Verify whether the triplets are Pythagorean?

AB =    AC2-AB2-BC2 =

BC= 

AC=     click

OR

n   =          r2-p2-n2 =
              (smallest number)

x   =     p (middle number) :
(difference in middle & smallest no.)

y   =     r (largest number) :
(difference in largest & middle no.)

            click           

Find out  the Pythagorean integer sub-triplets for given y :
( any integer starting from1)

y   =  Start value of n=
n   =    click

(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'+(2y2+2yx)1/2 and n'=y'+(2y2+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.)

y   =             

pair no.   = 

n =                

x   =                click

smallest number:

middle   number: 

largest  number:  

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.

n n' dn:
x , x' dx:,
y , y' dy:,
total , total     ,
principal*( p) <--principal* (q)     
Angle of point (θ) <--x'=0 for n'=total/2^1/2

Pythagorean Relatively Prime (test for prime no.) Triplets (click here to find GCD of 2 or more numbers)

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.