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.

1 Comment

Filed under Crypto, Geekhood, Math

One response to “It’s official: They’re prime

  1. I’ve never thought about modeling the number of digits but you’ve done it very well.

    This makes me wonder something, as you may know the prize for this first prime number of more than 10,000,000 has gone, so the first prime of 100,000,000 digits is now looking too far, but: wich one of the Mersenne numbers is the first that has more than 100,000,000 digits?

    It would be good to “refine” your model🙂