13th roots: Alexis Lemaire bests his own world record

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As widely reported in the on-line media ([AFP via Yahoo!], [Daily Telegraph (via NY Sun)]), “mathlete” Alexis Lemaire set a world record on Tuesday 11 Dec 2007 by computing the 13th root of a randomly generated 200 digit number in 70.2 seconds.  This improved on the previous world record of 72.4 seconds which he set in November 2007.  (“The first digit is easy and so is the last, but the middle ones are very hard”, Lemaire was quoted as saying in November.)

These times are measured starting when the number is first displayed, ending when the subject has finished writing down their answer.  In 2004, Lemaire moved on from the “easy” task of doing 13th roots of 100 digit numbers after setting the world record of just under 4 seconds; at that point, the challenge is no longer computational, but simply the speed with which one can write down the digits of the answer.

A BBC News article quoted Lemaire in July 2007:

When I think of numbers sometimes I see a movie, sometimes sentences. I can translate the numbers into words. This is very important for me. The art is to convert memory chunks into some kind of structure.

I see images, phrases, actions. It’s very tactile, sensitive. I have these associations between places and numbers. Some places are imaginary, I try to vary so I don’t confuse the numbers. It’s important to memorise. I have to be precise. [BBC News]

Just how does one compute 13th roots of large numbers in one’s head?  Two key components of the process:

  • Know your logarithms.  Memorizing the common logarithms of integers up to 125, combined with skill in factoring, suffices to deduce the first five digits of the 13th root of a 100 digit number.
  • Know your number theory.  13 x 77 =1001;  so if X is the 13th root of Y, then X is the 1001st root of Y^{77}.  But in general X^{1001} = X \pmod{10000} [* see footnote], so we can find the right-most significant digits of X by computing the 77th power of the right-most significant digits of Y.

The 13th root website has some discussion of the hows (and even the whys!) of doing such mental tasks, and delineates the ground rules to be followed when attempting to break the record.

As a challenge to our readers:  Find the 13th root of

960612190587317046682934201250686978925

0241422711135170561582869537947978835550

7267642189635009765625

and (as a bonus) explain the significance of the root you find. (No fair using Maple, Mathematica, or other CAS as an aid. But I’ll allow you to use paper and pencil, and a logarithm table. Can you get the answer in 4 seconds?)

* This follows from Euler’s Theorem. Note that 10000 = 16 x 625, and \phi(16)=8, and \phi(625)=500. Thus X^{1000} = (X^{125})^8 = 1 \pmod{16}, and X^{1000} = (X^{2})^{500} = 1 \pmod{625}, and hence X^{1000} = 1 \pmod{10000}.

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