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Catalan-Mersenne Number

Catalan (1876, 1891) noted that the sequence of Mersenne numbers 2^2-1=3, 2^3-1=7, and 2^7-1=127, and 2^(127)-1=170141183460469231731687303715884105727 (OEIS A007013) were all prime (Dickson 2005, p. 22). Therefore, the numbers defined by

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

with c_0=2 are known in this work as Catalan-Mersenne numbers.

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).

The Catalan-Mersenne numbers are a subset of the double Mersenne numbers.

See also

Double Mersenne Number, Mersenne Number, Mersenne Prime

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References

Catalan, E. Sur la théorie des nombres premiers. Turin, p. 11, 1876.Catalan, E. Théorie des nombres. p. 376, 1891.Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, 2005.Noll, L. K. "Prime Numbers, Mersenne Primes, Perfect Numbers, Etc." http://www.isthe.com/chongo/tech/math/prime/.Sierpiński, W. A Selection of Problems in the Theory of Numbers. New York: Macmillan, p. 91, 1964.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 81 1996.Sloane, N. J. A. Sequence A007013/M0866 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Catalan-Mersenne Number

Cite this as:

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

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