
Find the only Pythagorean triplet, {a, b, c}, for which a + b + c = 1000

I immediately Googled on how to find Pythagorean triplet and here it goes
Take a = m2 - n2
b = 2mn
c = m2 +n2
Hence a2+b2 = c2 and we get a triplet

well this method gives an idea how to form new triplets but not to find a triplet containing a given number

I was really interested in this

if a2 + 2ax = some square number .say y2 then we get a triplet ( x, y, x+a )
so for a given 'y' we have to find 'x' and 'a' to get the triplet
a2 = y2 – 2ax
Clearly ‘a’ should be lesser than square root o
Steps :
- Divide y2 by 'a'
- Subtract 'a' from it
- You shd get an EVEN number else choose another 'a' and goto 1
so ‘a’ can take value { 1 2 3 4 }
substituting a = 1 or 3 or 4 we get fractional values of ‘x’ hence we omit them
hence taking a=2 we get x=8
hence the triplet is (8, 6, 8+2) i.e. ( 8, 6, 10)

hence we can generate any triplet for a given number by choosing proper value for 'a'
Furthur observations :

2) Special case : consider y2= 4 or 1 . there are no values for 'a' in this case as square root are 2 and 1. Hence we cannot have a triplet containing 1 or 2
3) The above statement also proves that "There cannot be a right angled triangle with integer sides and any one of sides equal to 1 or 2"

to be continued

Yenda ipadi ... Yethavathu puiyara madiri post pannuda...:-):-)
ReplyDeletelol :) what dint u get da ??
ReplyDeletevery interesting post...i lik ur curiosity and innovative thinking..keep rocking..
ReplyDelete@shibani: Thanks a lot...!! :) :)
ReplyDelete