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The cube root of 5000 is approx 17.1 So the numbers 1 to 17 have cubes which are smaller than 5000 that is, there are 17 such numbers.
Probably infinite. The smallest is known as Ramanujan's number, 1729 using cubes of 1 and 12, and of 9 and10. Read more about the genius Ramanujan in Wikipaedia.
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