Of course what you are really wanting to know is how my spreadsheet models worked out for predicting the number of digits in these primes. First, the data:
- Number of digits actually in : 11,185,272
- Number of digits actually in : 12,978,189
My exponential model () was, unsurprisingly, way off — predicting a digit count of over 24.2 million for and over 35.8 million for . But the sixth-degree polynomial — printed on the scatterplot at the post linked to above — was… well, see for yourself:
- Number of digits predicted by 6th-degree polynomial model for : 11,819,349
- Number of digits predicted by 6th-degree polynomial model for : 13,056,236
So my model was off by 634,077 digits — about 6% error — for . But the difference was only 78,047 digits for , which is only about 0.6% error. That’s not too bad, if you asked me.
There’s only one piece of bad news that prevents me from publishing this amazing digit-count predicting device, and you can spot it in the graph of the model:
So evidently the number of digits in will max out around and then the digit count will begin to decrease, until somebody discovers , which will actually have no digits whatsoever. Um… no.