13th root 13th root Extracting the 13th root of a number is a famous category for the mental calculation world records. The challenge consists of being given a large number (possibly over 100 digits) and asked to return the number that, when taken to the 13th power, equals the given number. For example, the 13th root of 8,192 is 2 and the 13th root of 96,889,010,407 is 7. Extracting the 13th root has certain properties. One is that the 13th root of a number is much smaller: a 13th root will have approximately 1/13th the number of digits. Thus, the 13th root
Square root of 2 The number can also be expressed by taking the Taylor series of a trigonometric function. For example, the series for gives The Taylor series of with and using the double factorial gives The convergence of this series can be accelerated with an Euler transform, producing It is not known whether can be represented with a BBP-type formula. BBP-type formulas are known for and , however. The number can be represented by an infinite series of Egyptian fractions, with denominators defined by 2th terms of a Fibonacci-like recurrence relation a(n)=34a(n-1)-a(n-2), a(0)=0, a(1)=6. The square root of two has the following continued