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The amazing quartet! June 2, 2009

Posted by Ragesh G R in Physics and Maths, Uncategorized.

Long ago, in school, when I was solving an arithmetic problem, I observed that 6^3 = 5^3 + 4^3 + 3^3.  Interesting enough, I thought, we have atleast one set of such 4 integers x,y,z,w such that w^3 = z^3 + y^3 + x^3.

Now they were all adjacent/contiguous integers. Great! I thought.

Now if that was not interesting enough, the triplet on the right hand side of the equation is a pythagorean triplet! Wow! Great!

So we have a  set of quartets {x,y,z,w} which are even more exclusive than the Pythagorean triplets, such that w^3 = z^3 + y^3 + x^3 AND z^2 = y^2 + x^2 (That is to say x,y,z are Pythagorean triplets). Beautiful is n’t it?

To put it simply, we have some perfect cubes which can be expressed as the sum of the cubes of a pythagorean triplet.

Well how frequent are these kind of quartets ? Well an easy answer would be {30, 40, 50, 60}.

But the speciality with {3,4,5,6} is that all 4 are adjecent integers. So they are unique, because, all other quartets will spread out and won’t be contiguous integers.!

Disclaimer: Some of you may have known/observed this already.



1. mithblogsin - June 4, 2009

The first time I came across the quartet was when i was in class 11, not when solving an arithmetic problem, but when i was asked to write a program for Armstrong number.I did not know what an Armstrong number was then! These quartets have always fascinated me and the cubes n the squares are so beautiful..I can never stop admiring the intricacy involved in mathematics! Its beautiful.A great post after a long time! Cool observation !

Ragesh G R - June 5, 2009

He he thanks dude! cool! good to know you too observed them! yay!

2. Jithin K.Rajeev - June 5, 2009

That was an interesting observation!

Ragesh G R - June 5, 2009

Thanks dude!

3. BOB - June 7, 2009

Einstein’s discovery for the year 2k9
Congrats.. Hope you dint shout Eureka from bath tub 🙂

Ragesh G R - June 7, 2009

LOL! he he , thats coz i didnt discover it in the bathtub! LOL!

Ragesh G R - June 7, 2009

btw the discovery was in 1997! he he , published in 2009 LOL!

BOB - June 13, 2009

this is comment is like. ‘matter happened in 1997 and kid born in 2009’.. lol rofl 😀

Ragesh G R - June 13, 2009

@BOB: LOL! Ya ! a gestation period of 12 years rather than 12 months LOL! , btw I am not able to “Reply” to ur comment

Ragesh G R - June 13, 2009

Cos I think the max depth has bee reached in the comment thread hehe

4. mechanicmuthu - August 2, 2009

hav u wondered abt the Fermat’s (pronounced as Fermaa’s) last theorem.. It was kin of beautiful. and there is a whole audiobook in the torrrents about tht thing..

catch me at facebook

Ragesh G R - August 3, 2009

As a matter of fact, yes I have 🙂

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