An unsuspecting friend recently reported that her five-year-old son is a fan of big numbers in general, and one number in particular. His favorite number is One Googol and One Hundred (or 10100 + 100), which I will here on out refer to as the number “Samuel” to keep my sentences to a reasonable length. He wanted to be able to count to Samuel in the same way that you might count to other numbers by skipping a regular amount, such as counting to 1000 by tens.
As I started thinking through the various options, I realized that the answer depended on the prime factorization of Samuel, since any number by which you could count to Samuel would have to be a divisor of Samuel. One obvious divisor of Samuel is 100. Counting to Samuel by 100’s is certainly not feasible, but counting to Samuel by “Samuel divided by one hundred”s (or 1098 + 1 or one hundred untrigintillion and one) is, and that was my first suggestion. But then I wondered if the full factorization of Samuel would give other options. Factoring 100-digit numbers isn’t something that I know how to get my computer to do, but fortunately Wolfram Alpha was up to the task.
The result revealed that there are actually quite a handful of options for counting to Samuel in a reasonable amount of time, so I thought I would present them here, just for the joy of looking at really big numbers written out in words. These are all the numbers you can count to Samuel by in fewer than a thousand steps, sorted by and labeled by number of steps: