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Cubic AND Quartic Challenge (Posted on 2004-02-06) Difficulty: 4 of 5
What is the smallest positive integer that is the sum of two different pairs of (non-zero, positive) cubes?

What is the smallest positive integer that is the sum of two different pairs of integers raised to the 4th power? and how did you find it?

In other words what is the smallest x such that:
x = a^4 + b^4 = c^4 + d^4
(where x, a, b, c, and d are all different, non-zero, positive integers)?

Are you able to determine the answer without looking it up on the internet?

See The Solution Submitted by SilverKnight    
Rating: 2.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Question | Comment 7 of 15 |
(In reply to Question by Penny)

I've thought the same thing.... I'm guessing he might have been able to do the cubic in his head (although with some time for thought).

Possibly, he previously remembered many cubes in his mind (as we typically remember 169, 196, 225, 256, ...) and was able to consider a bunch. The solution has cubes no higher than 12^3, and so could be 'brute forced' relatively easily. And this, I think, is how the story goes. I believe it ends with Ramanujan asking about the 4th power, but he didn't have the answer handy....

I may be misremembering the story.

Either way, I seriously doubt a person, even he, could have done the quartic challenge in his head.
  Posted by SilverKnight on 2004-02-06 13:50:01

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