Thread: My Apologies
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Old 2010-08-02, 03:08   #70
lavalamp
 
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Oct 2007
Manchester, UK

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There is a little table on here:
http://www.mersenne.org/various/math.php

I'll reproduce it here slightly edited:
Code:
        Exponents     Bit
          up to      depth
        ---------    -----
         3960000      60
         5160000      61
         6515000      62
         8250000      63
        13380000      64
        23390000      65
        29690000      66
        37800000      67
        47450000      68
        58520000      69
        75670000      70
        96830000      71
This relationship between log(exponent) and bit depth is roughly linear, with a bit of a kink around 64 and 65 bits.

If you plot the log of the exponent against the bit depth (log to base 2 of the trial factor to depth), you get a nice straight line from a bit depth of 65 onwards.

I added a trend line to that and the equation was this:

ln(E) = 0.2346*D + 1.7171

Where E is the exponent and D is the bit depth.

So using this, it's tentatively possible to extend the table, but whether or not this linear relationship should continue I don't know. Assuming it does, here's what the numbers would be:
Code:
        Exponents     Bit
          up to      depth
        ---------    -----
        23390000      65
        29690000      66
        37800000      67
        47450000      68
        58520000      69
        75670000      70
        96830000      71
       120640000      72
       152530000      73
       192860000      74
       243860000      75
       308340000      76
       389860000      77
       492940000      78
       623280000      79
       788070000      80
       996440000      81
      1259900000      82
      1593020000      83
      2014220000      84
      2546790000      85
      3220170000      86
      4071590000      87
So it seems as though Operation Billion Digits should be taking numbers to 87 bits, and currently there are 10 (soon to be 13) exponents that are at 81 bits.

Also, the first lot of 100 million digit candidates should be trial factored to 77 bits.

Once again, I do not know if this linear relationship between exponent and trial factor depth should continue. So take it with a pinch of salt until someone who actually knows something about it makes a comment.

Last fiddled with by lavalamp on 2010-08-02 at 03:10
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