Mathematical pattern comparison software

Hello, I need a program that is capable of calculating very large numbers. It needs to be able to calculate in the 100 billion digit range. It needs to be able to do basic calculator functions such as Addition, subtraction multiplication, division, factorials and modulus function. I would also like to have the ability to create large numbers and store them in text documents. The main goal of the program is to compare a various numbers that is inputted to multiple list of numbers to find patterns in their results. We keep track of all the patterns. Eventually the patterns will begin to dissipate and cancel out previous results. When we get to the end there will either be no more values to compare against or some values that will negate all the other results.

While I am able to do this on smaller inputs I need to do the same on a far larger examples for a paper I am writing. I will be including a guide of the processes we run on these numbers. I will gladly answer any questions you may have and if the project and paper go well I will look to do some future projects I have planned related to this project.