How can you handle absurdly large numbers?

Question

There are some scenarios where programmers need or want to find grossly large numbers. These are often so large that they defy the programmer's comprehension. I'm talking about things like the largest known prime number (with 12978189 digits) and the recently calculated 10 trillion digits of pi.

How can you create a program that handles these? This far exceeds an integer, a long, a double, a BigInteger, a BigDecimal, or anything of the sort. How do these kinds of programs for discovering these numbers get created? How can you even store them in memory when no appropriate datatypes exist, and they would likely consume gigabytes of data each?

Answer 1

To address your specific examples:

• A 12 million digit integer isn't terribly large for a typical "large integer" class to handle. This should be able to be stored in memory.

• To store 10 trillion digits of π, you could use a disk file and memory-map it. You'll need a 64 bit OS and application, but you can simply create a 10 terabyte file on disk (you'll probably need a few disks and a filesystem like ZFS that can store it across disks), and map it into CPU address space. The algorithms that calculate π (such as BBP) conveniently calculate one hex digit at a time which fits well into half a byte of memory.

Answer 2

The (abstract) answer is to write algorithms using the machine's native types that produce the results you want. For instance, when you do addition by hand on paper of two very large integers, the biggest actual calculation you need is only 9+9+1 (nine plus nine plus one for the carry). Of course you need paper large enough to write the two numbers down in the first place and the answer down as well. So as long as the two numbers and the answer can be stored in a computer's harddisk (the paper), an algorithm can be written that does it with variables that only need a value up to 19; so even a char variable is more than capable of handling this let alone an int variable.

The (concrete) answer is that really good programmers have already done this and there even FOSS libraries for it. One good one is the GNU Project's GMP library which has loads of functions to handle arbitrary size integer arithmetic and arbitrary precision floating point arithmetic. So as long as your computer can store the information needing during the calculation, it can be done. You'll need to invest the time to read the documentation of course.