A152625 - OEIS

%I #13 Aug 10 2019 04:54:26

%S 197,223,227,229,257,263,283,311,317,379,383,389,457,461,463,467,521,

%T 541,569,607,661,701,751,773,787,839,859,863,881,887,907,971,991,1051,

%U 1061,1091,1153,1163,1171,1181,1213,1217,1277,1283,1301,1319,1321,1373

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

%C 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.

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

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

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

%H Amiram Eldar, <a href="/A152625/b152625.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatsLastTheorem.html">Fermat's Last Theorem</a>

%e 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.

%p 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:

%t 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 *)

%K nonn

%O 1,1

%A _Michel Lagneau_, Apr 04 2010