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Catalan-Mersenne Number

The name Catalan-Mersenne numbers is used in this work for the numbers defined by

c_n=2^(c_(n-1))-1

for positive integer n, with c_0=2.

Catalan (1876, p. 96) considered iterating the Mersenne number construction, noting the prime values 2^2-1=3, 2^3-1=7, and 2^7-1=127, and accepting Lucas's assertion that 2^(127)-1=170141183460469231731687303715884105727 is also prime (OEIS A007013; Dickson 2005, p. 22).

It is not known if c_5 is prime, but it is known that it has no prime factor less than 10^(51) (Noll; private correspondence with C. K. Caldwell, Aug. 10, 2003).

c_5 appeared in the Futurama movie The Beast with a Billion Backs (2008).

Every Catalan-Mersenne number c_n with n>=2 is a double Mersenne number, since c_n=2^(2^(c_(n-2))-1)-1. The initial terms c_0=2 and c_1=3 are exceptions.

See also

Double Mersenne Number, Mersenne Number, Mersenne Prime

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References

Catalan, E. Footnote to question 92 in "Questions proposées." Nouv. Corresp. Math. 2, 96, 1876. https://gdz.sub.uni-goettingen.de/download/pdf/PPN598948236_0002/LOG_0029.pdf.Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, 2005.Noll, L. C. "Landon Curt Noll's Prime Pages." https://www.isthe.com/chongo/tech/math/prime/.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 81, 1996.Sierpiński, W. A Selection of Problems in the Theory of Numbers. New York: Macmillan, p. 91, 1964.Sloane, N. J. A. Sequence A007013/M0866 in "The On-Line Encyclopedia of Integer Sequences."

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Catalan-Mersenne Number

Cite this as:

Weisstein, Eric W. "Catalan-Mersenne Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Catalan-MersenneNumber.html

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