What is the smallest positive integer that is the sum of two different pairs of (non-zero, positive) cubes?

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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)?

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

(In reply to

LOL by nikki)

And yet... someone answered without giving an explanation as to how he found the answer. :-( (At least he could/should have have indicated that he didn't have a way to prove or brute force it.)

BTW, *MY* solution would have been to write a program (similar to Charlie's) to brute force the answer. But I would very much like to see a less 'brute force' approach if someone finds a good way to 'cull' the solution domain--which I think is possible. And *could* be done manually (if one doesn't mind doing a few dozen/hundred long-multiplications, 4th power).