What is the second smallest integer with integral third, fourth and fifth roots?

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Problem

What is the second smallest integer with integral third, fourth and
fifth roots?  (1 is obviously the smallest.)

I solved this with a short C program, but was wondering if it could be
solved algebraically?

-- Bryan Moore

Show solution

The answer is 2^60 = 1152921504606846976.  Indeed, consider the prime
decomposition of x as p1^a1 . ... . pr^ar.  All the exponents have to
be multiples of 3, 4 and 5 at the same time (otherwise the
corresponding roots are not integers), so since 3, 4 and 5 are mutually
prime, all exponents are multiples of 60. The smallest such integer is
2^60, followed by 3^60, 4^60, etc...

-- Denis Auroux <auroux@clipper.ens.fr>