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A158477 Primes p with property that Q(p) = p^32+2^32 is prime.

%I A158477 #24 Jul 08 2020 03:25:18
%S A158477 29,59,101,103,109,181,199,379,769,881,919,977,1097,1213,1303,1583,
%T A158477 2099,2113,2441,2521,2617,2777,3067,3739,4133,4289,4519,4931,5039,
%U A158477 5113,5227,5417,5743,5783,6143,6373,6691,8053,8209,8287,8513,9109,9203,9689,9787,9923,9941
%N A158477 Primes p with property that Q(p) = p^32+2^32 is prime.
%C A158477 1) Q=(p^16)^2+(2^16)^2 only for Q=4k+1 because of Fermat/Euler/Lagrange theorem concerning prime as sum of two squares.
%C A158477 2) It is conjectured that sequence a(n) is infinite.
%C A158477 3) Note the twin prime: a(3)=101, a(4)=103.
%C A158477 The next set of twins is a(101)=30557, a(102)=30559. - _Robert Israel_, Apr 05 2016
%D A158477 Richard E. Crandall, Carl Pomerance, Prime Numbers: A Computational Perspective, Springer 2001.
%D A158477 Leonard E. Dickson, History of the Theory of Numbers, Dover Pub. Inc., 2005.
%H A158477 Robert Israel, <a href="/A158477/b158477.txt">Table of n, a(n) for n = 1..10000</a>
%F A158477 n^32+2^32 and n to be prime.
%e A158477 p=3: 3^32+2^32=1853024483819137 = 1153 x 1607133116929 no prime;
%e A158477 also for following primes p=5, 7, 11, 13, 17, 19, 23: Q(p) no prime;
%e A158477 p=29: 29^32+2^32=62623297589448778360828428329074752313100292737 is prime => a(1)=29.
%p A158477 select(t -> isprime(t) and isprime(t^32 + 2^32), [seq(i,i=3..10000,2)]); # _Robert Israel_, Apr 05 2016
%t A158477 With[{c=2^32},Select[Prime[Range[1300]],PrimeQ[#^32+c]&]] (* _Harvey P. Dale_, May 04 2018 *)
%o A158477 (PARI) isA158477(n) = isprime(n) && isprime(n^32+4294967296) \\ _Michael B. Porter_, Dec 17 2009
%o A158477 (PARI) lista(nn) = forprime(p=3, nn, if(ispseudoprime(p^32+2^32), print1(p, ", "))); \\ _Altug Alkan_, Apr 05 2016
%Y A158477 Cf. A062324, A157764, A157950, A158361.
%K A158477 nonn
%O A158477 1,1
%A A158477 Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 20 2009