Determine the smallest possible positive integer P which is not a perfect seventh power, but in the decimal expansion of its seventh root, the decimal point is followed by at least ten consecutive zeroes.

My thought was that the number would be close to an actual seventh power and since we're looking for the smallest, use x^7 + 1.  Note that x^7 - 1 will give a number looking like (x-1).9999999...

Cranking some values using Excel of (x^7+1)^(1/7)-x

34^7 + 1 = 52523350145

(34^7+1)^(1/7) - 34 = 9.247e-11     or

52523350145 ^ (1/7) = 34.00000000009247

Posted by Bob Smith on 2006-04-05 14:44:59