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.

(n^7+1)^(1/7) - n < 10^-10
n^7+1 < (n + 10^-10)^7
n^7+1 < n^7 + 7n^6 * 10^-10 + ... + 7n * 10^-60 + 10^-70
1 < 7n^6 * 10^-10 + 21n^5 * 10^-20 + ... + 7n * 10^-60 + 10^-70
All but the first term appear to be of negligible magnitude, so I will remove them.

1 < 7n^6 * 10^-10
10^10 < 7n^6
n = ceiling( (10^10 /7)^(1/6) ) = 34

Therefore P = 34^7 + 1 = 52523350145
unless I have rounding errors.

Edit: parentheses added

Edited on April 5, 2006, 9:01 pm

Posted by Tristan on 2006-04-05 20:59:47