Pythagorean Triplets

Whether the TRIPLET is Pythagorean ? Put the 2 short numbers-M, N M-  and N-  Highest No. P- M+N+P:
In case of above TRIPLET, if the triplet is reducible ?                          GCD of M,N,P: Reduced :MNP&M+N+P
With x > y and x, y being whole numbers, put any value of x=   and y =  a= , b = , c= ;a+b+c=
the primitive triplets are x2-y2, 2xy, x2+y2    
Put start no. as 1 and give the number of iteration. Pythagorean no. iteration incre The series-- 1+3+7+17+41+99+239+........
triplets will be generated which when divided by 2 shall reduce Exa-(3,4,5)(20,21,29)(119,120,169)(696,697,985)

From 3rd no. every no=2*previous no. +pre-previous number.

to primitive form with 2 numbers as consecutive numbers.   for further work, click

In case of pythagorean triplets where 2 numbers are consecutive, the 3rd number T3has pattern. T3(1) = (12+4)/1 =5,T3(2)=(52+4)/1=29, T3(3)=(292+4)/5=169, T3(4) = (1692+4)/29 =985, so on with start up no. 5. Put 5 or 1 in start no., 4 in incre ,put the no.of iteration and click near c to get T(3).TRY with different start no. and incre as start no. -1 to find patterns.

*The radius of incircle of a pythagorean triangle is always a whole number.

*Gaussian Triplets-- If A, B, C are three Gaussian numbers such that A2+B2=C2, then A, B and C are called Gaussian triplets. Gaussian numbers are complex no. of the form a±bi where a, b are real numbers and i is a complex number. Exa--(1+4i)2 + (8-4i)2 = (7-4i)2

*In Pythagorean triplets, long side is always odd number, the other 2 sides-one even and the other odd. If the last digit of

2 short sides is (0,1),(0,9)--> long side is (1 or 9).

2 short sides is (3,4),(3,6),(7,4),(7,6)--> long side is (5).

2 short sides is (5,2),(5,8)--> long side is (3 or 7).

* If product of 2 short sides which are integers is divided by the long side which is also an integer and the result is an integer, then  the perpendicular from the right angle on hypotenuse will be an integer which does not happen.  The perpendicular divides the long side in the ratio, say a & b, then a = square of 1 shortside/ longside and b= square of other short side/ long side;

* Some of the irreducible Pythagorean triplets are (3,4,5),(5,12,13),(8,15,17),(7,25,29),(20,21,29),(12,35,37),(9,40,41),(28,45,53),(11,60,61),(16,63,65),(33,56,65),(48,55,73),(13,84,85),(36,77,85),(39,80,89),(20,99,101),(65,72,97),(15,112,113),(60,91,109),(45,108,117),(44,117,125),(17,144,145),(24,143,145),(88,105,137),(51,140,149),(19,180,181),(52,165,173) with total 12,30,40,56,70,84,90,126,132,144,154, 176,182,198,208,220,234,240,260, 270,286,306, 312,330 ,340,380,390 respectively in between 1-400. Those triplets in red have the same long side but different short sides. The long side can be a diameter of a circle and the 2 short sides are integer valued short sides meeting at a point on the circumference subtending a right angle. All other short sides meeting at a point on the circumference are not of integer value.

* Sum of Pythagorean triplets is always even number.


No. of Pythagorean Triplets

Sum of the triplets No. of Irreducible Triplets No. of Reducible  Triplets
1-100 7 10
101-200 7 19
201-300 7 24
301-400 6 25
1-400 27 78


References- Pythagorean triplets