Big numbers

Big numbers, known colloquially as bignums (adjectival form bignum, as in bignum arithmetic) are integers whose range exceeds those of machine registers. For example, most modern processors possess 64-bit registers which can be used to store integers up to 2⁶⁴-1. It is usually possible to add, subtract, multiply, or divide such integers in a single machine instruction. However, such machines possess no native implementation of arithmetic on numbers larger than this, nor any native means of representing them. In some applications, it might be necessary to work with numbers with hundreds or even thousands of digits.

Fixed versus dynamic bignums

There are, in principle, two kinds of bignum implementation. Suppose we know in advance the maximum size of the integers we might be working with. For example, in TREE1@SPOJ, we are asked to report the number of permutations which satisfy a certain property. There are only up to 30 elements, so we know that the answer will not exceed 30! = 265252859812191058636308480000000, which has 33 digits. It is not terribly difficult to implement the solution in such a way that no intermediate variable is ever larger than this. So we could, for example, use a string of length 33 to store all integers used in the computation of the answer (where numbers with fewer than 33 digits are padded with zeroes on the left), and treat all numbers as though they had 33 digits. This is probably the easiest type of bignum to implement.