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A103902 Mersenne primes p such that the Mersenne number M(p) = 2^p - 1 is composite.

%I A103902 #13 Feb 16 2025 08:32:56
%S A103902 8191,131071,524287,2147483647
%N A103902 Mersenne primes p such that the Mersenne number M(p) = 2^p - 1 is composite.
%C A103902 Only four terms are known.
%C A103902 The first four Mersenne primes (p=2^q-1 in A000668) are double Mersenne primes, i.e., in A103901. The next four yield a composite M(p) and therefore are in this sequence. The next larger Mersenne prime p = A000668(9) has already 19 digits and is much too large to enable us, as of today, to test the primality of 2^p-1 (which would require over 10^8 gigabytes just to be stored in binary). This explains that only 4 terms are known of this sequence and of A103901; for all the 30+ remaining members of A000668 it is not known whether they belong to A103901 or to this sequence A103902. - _M. F. Hasler_, Jan 21 2015
%D A103902 R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer-Verlag, NY, 2004, Sec. A3.
%D A103902 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford Univ. Press, 1954, p. 16.
%D A103902 P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, NY, 1996, Chap. 2, Sec. VII.
%H A103902 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/mersenne/index.html#unknown">Mersenne Primes: Conjectures and Unsolved Problems</a>
%H A103902 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DoubleMersenneNumber.html">Double Mersenne Number</a>
%H A103902 Wikipedia, <a href="http://www.wikipedia.org/wiki/Mersenne_prime">Mersenne prime</a>
%e A103902 M(13) = 8191 is a Mersenne prime and M(1891) is composite, so 1891 is a member.
%o A103902 (PARI) is(n)=isprime(2^n-1) && !isprime(2^(2^n-1)-1) \\ _Charles R Greathouse IV_, Jan 24 2015
%Y A103902 Cf. A000043, A000668, A001348, A077585, A077586, A103901.
%K A103902 hard,nonn
%O A103902 1,1
%A A103902 _Jonathan Sondow_, Feb 20 2005