Tag Archives: prime numbers

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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Filed under Crypto, Geekhood, Math

Wednesday link-fest

You know, there’s some good stuff showing up in my RSS reader once I get a chance to read it:

  • There’s a 21-page paper titled “Are There Infinitely Many Primes?” over at arXiv. How do you write 21 pages on a question that was answered “yes” about 2500 years ago? You’ll have to go see for yourself.
  • xkcd turns the Turing Test around.
  • IHE has this article on dual enrollment (high school students taking college courses) and its benefits. I agree. I’ve been involved with a dual-enrollment program at my college, and I’m definitely preferring this approach over taking a so-called AP course taught and designed outside the auspices of a college that may or may not prepare students well for actual college courses.
  • Dana Huff is wondering whether there are programs out there that will donate laptops to teachers. There’s this program from the One Laptop Per Child project, but I’ve not seen a similar program for teachers looking for “grown-up” laptops. Anybody able to help her out?
  • Homeschool2.0 has a cartoon to share about the socialization of homeschooled kids.
  • Jackie at Continuities joins me in my skepticism about digital natives.

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Filed under Education, Educational technology, Math, School choice, Teaching, Technology