Question

I am writing a program where I need to know only the first k (k can be anywhere between 1-5) numbers of another big number which can be represented as n^n where n is a very large number.

Currently I am actually calculating n^n and then parsing it as a string. I wonder if there is a better more fast method exists.

Answer 1

There are two possibilities.

If you want the first k leading digits (as in: the leading digit of 12345 is 1), then you can use the fact that

n^n = 10^(n*Log10(n))

so you compute the fractional part f of n*Log10(n), and then the first k digits of 10^f will be your result. This works for numbers up to about 10^10 before round-off errors start kicking in if you use double precision. For example, for n = 2^20, f = 0.57466709..., 10^f = 3.755494... so your first 5 digits are 37554. For n = 4, f = 0.4082..., 10^f = 2.56 so your first digit is 2.

If you want the first k trailing digits (as in: the trailing digit of 12345 is 5), then you can use modular arithmetic. I would use the squaring trick:

factor = n mod 10^k
result = 1
while (n != 0) 
    if (n is odd) then result = (result * factor) mod 10^k
    factor = (factor * factor) mod 10^k
    n >>= 1

Taking n=2^20 as an example again, we find that result = 88576. For n=4, we have factor = 1, 4, 6 and result = 1, 1, 6 so the answer is 6.