A152625 Prime numbers p such that 2p+1, 4p+1, 8p+1, 10p+1, 14p+1 and 16p+1 are all composite numbers.

197, 223, 227, 229, 257, 263, 283, 311, 317, 379, 383, 389, 457, 461, 463, 467, 521, 541, 569, 607, 661, 701, 751, 773, 787, 839, 859, 863, 881, 887, 907, 971, 991, 1051, 1061, 1091, 1153, 1163, 1171, 1181, 1213, 1217, 1277, 1283, 1301, 1319, 1321, 1373

Offset

1,1

Comments

Related to Legendre's contribution to Fermat's last theorem: the first case of Fermat's last theorem is true only if the Diophantine equation x^n + y^n = z^n has integer solutions x,y,z where n is prime such that gcd(n, xyz) = 1, then n >= 197.

References

A. E. Bojarincev, Asymptotic expressions for the n-th composite number, Univ. Mat. Zap. 6:21-43 (1967). - In Russian.

J. M. De Koninck, Ces nombres qui nous fascinent, Ellipses, 2008, p. 59.

Sampson, J.H. "Sophie Germain and the theory of numbers," Arch. Hist. Exact Sci. 41 (1990), 157-161.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Fermat's Last Theorem

Example

With p=197 we obtain the composite numbers 2p+1 = 5*79, 4p+1 = 3*263, 8p+1 = 19*83, 10p+1 = 27*73, 14p+1 = 31*89 and 16p+1 = 3*1051.

Maple

for p from 1 to 2000 do: if type(p, prime)=true and type(2*p+1, prime)=false and type(4*p+1, prime)=false and type(8*p+1, prime)=false and type(10*p+1, prime)=false and type(14*p+1, prime)=false and type(16*p+1, prime)=false then print(p):else fi:od:

Mathematica

aQ[p_] := PrimeQ[p] && AllTrue[{2 p + 1, 4 p + 1, 8 p + 1, 10 p + 1, 14 p + 1, 16 p + 1}, CompositeQ]; Select[Range[1400], aQ] (* Amiram Eldar, Aug 10 2019 *)

Crossrefs

Sequence in context: A296895 A171383 A182572 * A082246 A159809 A345551

Adjacent sequences: A152622 A152623 A152624 * A152626 A152627 A152628

Keyword

nonn

Author

Michel Lagneau, Apr 04 2010

Status

approved