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Double Mersenne number

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A double Mersenne number is a number where the exponent is also a Mersenne number and usually a Mersenne prime. These are generally denoted as MMp, MMp, or MM(p) and refer to [math]\displaystyle{ 2^{2^p-1}-1 }[/math]. Early on it was thought that if M(p) was prime so too was MM(p).

This holds true for the first 4 terms:

  • MM(2) = [math]\displaystyle{ 2^3-1 }[/math] = 7, known prime since antiquity
  • MM(3) = [math]\displaystyle{ 2^7-1 }[/math] = 127, known prime since antiquity
  • MM(5) = [math]\displaystyle{ 2^{31}-1 }[/math] = 2147483647, proven prime by Leonhard Euler in 1772
  • MM(7) = [math]\displaystyle{ 2^{127}-1 }[/math] = 170141183460469231731687303715884105727, proven prime by Édouard Lucas in 1876

However this does not hold true for next 4 terms:

  • MM(13) has a factor, 338193759479, found in 1976
  • MM(17) has a factor, 231733529, found in 1957
  • MM(19) has a factor, 62914441, found in 1957
  • MM(31) has a factor, 295257526626031, found in 1983

All further numbers are MM(61), MM(89), MM(107), MM(127), etc. are unknown. (It is worth noting that, there was a gap of over 80 years between the proving of MM(7) and the finding of factors for MM(17) & MM(19). The factors were found by Raphael M. Robinson with a computer.) These all are so very large that there is no hope of running a Lucas-Lehmer test on them any time in the forseeable future. Efforts are under way attempting to find factors for these numbers. Currently MM(61) and MM(127) are getting the most attention.

MM(61) or [math]\displaystyle{ 2^{2305843009213693951}-1 }[/math] is 694 127 911 065 419 642 digits long. Tony Forbes lead an effort to find a factor for this number. The search has included all k values up to 1,167,025,860,000,000, with many beyond that as well (as of Feb 2011).

George Woltman wrote a GPU program mmff that can factor double Mersennes. Luigi Morelli is co-ordinating the effort to factor these numbers.

More about MM(127) below.

Catalan Sequence

A subset of double Mersennes is the Catalan-Mersenne sequence (sometimes called simply the Catalan sequence). The sequence was first noted by Eugéne Catalan in 1876 after Lucas' finding M(127) to be prime.

The sequence can be written several ways, as noted in the table below:

element #
Mersenne function Mersenne element # value
C0 2 2
C1 [math]\displaystyle{ 2^{C_0}{-}1 }[/math] M(2) M1 3
C2 [math]\displaystyle{ 2^{C_1}{-}1 }[/math] M(3) or
M2 7
C3 [math]\displaystyle{ 2^{C_2}{-}1 }[/math] M(7) or
M4 127
C4 [math]\displaystyle{ 2^{C_3}{-}1 }[/math] M(127) or
M12 170141183460469231731687303715884105727
C5 [math]\displaystyle{ 2^{C_4}{-}1 }[/math] M(170141183460469231731687303715884105727) or
M??, status unknown [math]\displaystyle{ \approx1*10^{(5.12*10^{30})} }[/math]

This was the sequence that Lucas was testing when he proved M(127) to be prime. While the sequence does hold true for the first few elements, this likely to be a case of the "Strong law of small numbers". There have been efforts to again test this trend, by finding a factor for C5. If a factor is found M(M(127)), then M(M(M(127)) etc. must be composite. Landon Curt Noll has trial factored this number up to a k value of at least 3,000,000,000,000 (i.e. there is no factor smaller than [math]\displaystyle{ 5*10^{51} }[/math]), a bit level over 169.4. The current version of Prime95 cannot handle numbers this large, nor can mfaktc.

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