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 A152625 Prime numbers p such that 2p+1, 4p+1, 8p+1, 10p+1, 14p+1 and 16p+1 are all composite numbers. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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, aQ] (* Amiram Eldar, Aug 10 2019 *) CROSSREFS Sequence in context: A296895 A171383 A182572 * A082246 A159809 A051371 Adjacent sequences:  A152622 A152623 A152624 * A152626 A152627 A152628 KEYWORD nonn AUTHOR Michel Lagneau, Apr 04 2010 STATUS approved

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Last modified April 1 08:24 EDT 2020. Contains 333159 sequences. (Running on oeis4.)