# Tag Archives: prime

## It’s official: They’re prime

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.

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## It’s a prime! And another prime!

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!

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## New Mersenne prime discovered?

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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