I have calculated the first 200,000 decimal digits of Sophomore's dream, a mathematical constant described by the following formula:
I am probably the first and only person on earth to have calculated that many decimal digits of Sophomore's dream. Because calculation needs fast hardware and fast hardware is expensive, donations are greatly appreciated.
Since no closed form exists for Sophomore's dream, it is still very difficult to calculate a vast number of digits. The approach is, however, trivial.
For the 200,000 decimal digits that are available here, I effectively calculated:
This actually allows me to calculate a bit more than 200,000 decimal digits. But for precision's sake, I threw away some to be certain about the first 200,000 digits.
To avoid floating-point specifics, I worked with fractions that I divided later on. I also made use of the following equation that allows me to dismiss finding the LCM for every addition of two fractions:
That way, finding a vast number of decimal digits of Sophomore's dream is now only a matter of computational power. And you need lots of computational power to calculate Sophomore's dream! My personal digital computer managed to calculate the constant's first 200,000 decimal digits in about 40 hours. Without massive parallelization efforts, that would not have been possible. The subsequent division to get the decimal representation of the calculated super-fraction took about 30 hours. Even though computation becomes superexponentially difficult, I am currently planning to calculate Sophomore's dream's first 1,000,000 or more digits. Using CRIU, this lengthy process can be portioned out easily.
If you want to do this calculation too, this might be helpful for you if you want to use parallelization, and thus partial sums.