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a^{2} + b^{2} = c^{2}

I was going through the problems posted in Project Euler and was suddenly struck with this problem Pythagorean Triplet

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

Given any two arbitrary numbers ‘m’ and ‘n’

Take a = m^{2} - n^{2}

b = 2mn

c = m^{2} +n^{2}

Hence a^{2}+b^{2} = c^{2} 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 now consider the expansion (x + a)^{2} = x^{2}+ a^{2} +2ax

if a^{2} + 2ax = some square number .say y^{2} 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 ( x, y, x+a )y^{2} = a^{2} + 2ax = a ( a + 2x )

a^{2} = y^{2} – 2ax

Clearly ‘a’ should be lesser than square root of ‘y’

Steps :

- Divide y
^{2} by 'a' - Subtract 'a' from it
- You shd get an EVEN number else choose another 'a' and goto 1

For eg. take 36… Factors are – 1 2 3 4 6 9 12 18 36

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 : 1) 'a' is odd for a given odd square number and vice versa

2) Special case : consider y^{2}= 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"

I Googled on this topic and found nothing regarding this

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