The numbers believed to be the 45th and 46th Mersenne primes have been proven to be prime. The 45th Mersenne prime is \(2^{37156667} -1\) and the 46th is \(2^{43112609} – 1\).Full text of these numbers is here and here.

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 \(M_{45}\):
**11,185,272** - Number of digits actually in \(M_{46}\):
**12,978,189**

My exponential model (\(d = 0.5867 e^{0.3897 n}\)) was, unsurprisingly, way off — predicting a digit count of over 24.2 million for \(M_{45}\) and over 35.8 million for \(M_{46}\). 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 \(M_{45}\):
**11,819,349** - Number of digits predicted by 6th-degree polynomial model for \(M_{46}\):
**13,056,236**

So my model was off by 634,077 digits — about 6% error — for \(M_{45}\). But the difference was only 78,047 digits for \(M_{46}\), 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 \(M_{n}\) will max out around \(M_{49}\) and then the digit count will begin to decrease, until somebody discovers \(M_{55}\), which will actually have *no* digits whatsoever. Um… no.