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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September 11, 2008, 8:13 pm
By Robert Talbert
The number believed to be the 45th Mersenne prime has turned out actually to be a prime, according to GIMPS. The verification was completed on 6 September and announced on 7 September.
But in a fairly extraordinary turn of events, yet another number was submitted to the GIMPS servers as the next possible Mersenne prime on 6 September — and the initial verification shows that it is prime too! So we now have the 45th and 46th Mersenne primes discovered within two weeks of each other, which is incredible.
No word yet on the details of these primes. We’ll soon see who wins the Mersenne prime digit-guessing challenge. You can still play along with your own spreadsheet too!
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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August 26, 2008, 8:05 pm
By Robert Talbert
GIMPS is reporting that on 23 August a new Mersenne prime was reported to their server. Verification began today and should take about two weeks to complete. No word on what the prime was, how many digits, etc.
The last Mersenne prime discovered was \(2^{32,582,657}-1\), back in 2006 (blogged about here) and weighed in at a whopping 9,808,358 digits. Any bets on how big this new one is, if it’s really a prime? I’m guessing 10.5 million digits. Sounds like a good occasion for a nerd office pool.
Update: Isabel at God Plays Dice likes 14.5 million digits instead, and she’s actually using math and stuff to make that estimate instead of just shooting totally in the dark like I am.
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