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A119859 Lesser of the twin primes formed by 8x^4-1 and 8x^4+1 where x is a multiple of 3.

%I A119859 #16 Dec 24 2019 08:26:38
%S A119859 253124999,10871635967,14688294407,168573727367,196730062847,
%T A119859 248935679999,528593507207,759035204999,956311308287,1602486789767,
%U A119859 2451216826367,9613393373447,18132940558727,60600405623687
%N A119859 Lesser of the twin primes formed by 8x^4-1 and 8x^4+1 where x is a multiple of 3.
%C A119859 Theorem: 8x^4-1 and 8x^4+1 can both be prime iff x = 3m for some integer m. Proof: If x != 3m then x=3m+1 or x=3m+2. If x = 3m+1, then 8x^4+1 = 8(81*m^4 + 108*m^3 + 54*m^2 + 12*m)+8+1 = 3H for some H. If x = 3m+2, then 8x^4+1 = 8(81*m^4 + 216*m^3 + 216*m^2 + 96*m)+128+1 = 3H for some H. Since 8x^4+1 cannot be prime for x != 3m for all m, it follows that 8x^4-1 and 8x^4+1 can both be prime only if x = 3m for some m. A proof that this sequence is infinite would be good to have.
%H A119859 Amiram Eldar, <a href="/A119859/b119859.txt">Table of n, a(n) for n = 1..10000</a>
%e A119859 For x=75, 8x^4-1 = 253124999 prime, 8x^4+1 = 253125001 prime so 253124999 is the first entry.
%t A119859 Select[8*(3Range[600])^4,And@@PrimeQ[{#+1,#-1}]&]-1 (* _Harvey P. Dale_, Feb 14 2013 *)
%o A119859 (PARI) twin8k3(n) = {local(a,b,c,x); c=0; forstep(x=3,n,3, a=8*x^4-1; b=8*x^4+1; if(ispseudoprime(a)&ispseudoprime(b), c++; print1(a","); ); ); print(); print(c) }
%K A119859 nonn
%O A119859 1,1
%A A119859 _Cino Hilliard_, Jul 31 2006
%E A119859 Offset corrected by _Amiram Eldar_, Dec 24 2019