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
sum of the square of
two smaller sides of a right angled triangle equals square of the third side
sum of the square of
two rational numbers equals square of the third number. These numbers are
called Pythagorean triplets.
the triplets are Pythagorean?
(middle number) :
(difference in middle & smallest no.)
(difference in largest & middle no.)
Find out the Pythagorean integer
sub-triplets for given
( any integer starting from1)
Start value of n=
start value of 'n' or subsequent integer value)
(only pick up integer values)
Some Interesting Properties of
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
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
& 1st,3rd,5th,7th.. ..no. are with n/y
5/2,7/2,9/2,11/2...with common difference of 1)
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
10. For given y, there is a constant difference
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
and its odd multiples where x is an odd integer.
12. For each integer value of y, infinite number of triplets can
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.)
Next Level of triplets (explore with above largest number as the smaller
side of another right angled triangle in 3-D space)
(y, n, x chart at different y values)
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.
|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
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.