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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.