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

8191, 131071, 524287, 2147483647

Offset: 1

Author

Jonathan Sondow, Feb 20 2005

Keywords

• hard
• nonn

Comments

Only four terms are known.

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

Examples

			M(13) = 8191 is a Mersenne prime and M(1891) is composite, so 1891 is a member.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer-Verlag, NY, 2004, Sec. A3.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford Univ. Press, 1954, p. 16.
• P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, NY, 1996, Chap. 2, Sec. VII.

Links

• C. K. Caldwell, Mersenne Primes: Conjectures and Unsolved Problems
• Eric Weisstein's World of Mathematics, Double Mersenne Number
• Wikipedia, Mersenne prime

Crossrefs

Cf. A000043, A000668, A001348, A077585, A077586, A103901.

Programs

PARI

is(n)=isprime(2^n-1) && !isprime(2^(2^n-1)-1) \\ Charles R Greathouse IV, Jan 24 2015