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Double Mersenne number
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 MM_{p}, MMp, or MM(p) and refer to [math]\displaystyle{ 2^{2^p1}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{ 2^31 }[/math] = 7, known prime since antiquity
 MM(3) = [math]\displaystyle{ 2^71 }[/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 LucasLehmer 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 coordinating the effort to factor these numbers.
More about MM(127) below.
Catalan Sequence
A subset of double Mersennes is the CatalanMersenne 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:
Catalan element # 
Catalan formula 
Mersenne function  Mersenne element #  value 

C_{0}  2  2  
C_{1}  [math]\displaystyle{ 2^{C_0}{}1 }[/math]  M(2)  M_{1}  3 
C_{2}  [math]\displaystyle{ 2^{C_1}{}1 }[/math]  M(3) or M(M(2)) 
M_{2}  7 
C_{3}  [math]\displaystyle{ 2^{C_2}{}1 }[/math]  M(7) or M(M(M(2))) 
M_{4}  127 
C_{4}  [math]\displaystyle{ 2^{C_3}{}1 }[/math]  M(127) or M(M(M(M(2)))) 
M_{12}  170141183460469231731687303715884105727 
C_{5}  [math]\displaystyle{ 2^{C_4}{}1 }[/math]  M(170141183460469231731687303715884105727) or M(M(M(M(M(2))))) 
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 C_{5}. 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.
External links
 Double Mersennes Prime Search by Luigi Morelli
 Archive of Will Edgington's http://www.garlic.com/~wedgingt/MMPstats.txt on Double Mersennes (known factors, factoring effort, and more.)
 Tony Forbes' project http://sites.google.com/site/anthonydforbes/mm61.htm > unreachable
 MersenneForum subforum for Double Mersennes
General numbers 
Special numbers 

Prime numbers 
