Find the smallest natural number which can be written as the sum of three cubes in two different ways? Generalised Ramanujan Number


closed as off-topic by Namaste, C. Falcon, Shailesh, Zain Patel, JonMark Perry May 3 '17 at 5:41

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  • 2
    $\begingroup$ You find it. And if you can't, show us what you tried, and you'll get some help. $\endgroup$ – Alec May 2 '17 at 10:53
  • $\begingroup$ A copy past of your question on Google yield mathworld.wolfram.com/Hardy-RamanujanNumber.html $\endgroup$ – Zubzub May 2 '17 at 10:54

Consider the identity - $$(a+b+c)^3-24abc=(a+b-c)^3+(a-b+c)^3+(b+c-a)^3$$ and $$(a+b+c)^3+a^3+b^3+c^3=(a+b)^3+(a+c)^3+(b+c)^3+6abc$$ Put $a=2$,$b=3$,$c=6$ in the second identity so that the number becomes $1366=2^3+3^3+11^3=5^3+8^3+9^3$ (if the numbers are distinct then 1366 is the smallest number)

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    $\begingroup$ OEIS A024974 disagrees. $1009=1^3+2^3+10^3=4^3+6^3+9^3$. +1 for the useful identities though. $\endgroup$ – nickgard May 2 '17 at 13:31
  • $\begingroup$ Ok I was not aware of that $\endgroup$ – Angad May 2 '17 at 19:37

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