i was intrigued by this report in the Australian of a mathematical genius who can find the 13th root of 100-digit number in seconds. I was even more intrigued to learn that the number was “selected at random”. The odds against a randomly selected 100-digit number having an integer 13-th root would be a nice problem for a math genius to solve in seconds so I wonder what is meant by this.
Maybe you only have to derive the integer part, which, if my workings below are correct, is an eight-digit number lying in a fairly narrow range. Someone who knew the first three or four places of their log tables and was quick at interpolation could probably manage the feat in the time described.
Or maybe the number is selected from a list of 100-digit 13th powers. This would make life a little easier since you can get the last digit free or cheaply (with preparation, you could probably get a good handle on the last two digits).
I’m not planning on trying this at home, though.
fn1. Not being a math genius, I’d proceed as follows. The number lies between 10^99 and 10^100, and there are roughly 10^100 such numbers. The 13th root lies between 10^99/13 or about 10^7.63 and 10^100/13 or about 10^7.69 and there are about 10^6 integers in this range. So the odds against are 10^94 to 1, which might fairly be described as “long”. Time taken, about 150 seconds, which I guess counts as “in seconds” if you’re feeling generous.