September 17, 2008, 11:56 am

By Robert Talbert

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 \(M_{45}\):…

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August 28, 2008, 1:57 pm

By Robert Talbert

At the end of this post, I made a totally naive guess that the recently discovered candidate to be the \(M_{45}\), the 45th Mersenne prime, would have 10.5 million digits. There was absolutely no systematic basis for that guess, but I did suggest having an office pool for the number of digits, so what I lack in mathematical sophistication is made up for by my instinct for good nerd party games. On the other hand, Isabel at God Plays Dice predicted 14.5 million digits based on a number theoretic argument. Since I am merely a wannabe number theorist, I can’t compete with that sort of thing. But I *can* make up a mean Excel spreadsheet, so I figured I’d do a little data plotting and see what happened.

If you make a plot of the number of digits in \(M_n\), the nth Mersenne prime, going all the way back to antiquity, here’s what you get:

The horizontal axis is *n* and the vertical…

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