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A119860 Greater of the twin primes of the form 8*k^4 - 1 and 8*k^4 + 1 where k is a multiple of 3. 1

%I #10 Dec 26 2019 05:06:44

%S 253125001,10871635969,14688294409,168573727369,196730062849,

%T 248935680001,528593507209,759035205001,956311308289,1602486789769,

%U 2451216826369,9613393373449,18132940558729,60600405623689,142671521205001,178044790376449,261461826945289,290127048939649

%N Greater of the twin primes of the form 8*k^4 - 1 and 8*k^4 + 1 where k is a multiple of 3.

%C 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 Amiram Eldar, <a href="/A119860/b119860.txt">Table of n, a(n) for n = 1..10000</a>

%e For k = 75, 8*k^4 - 1 = 253124999 is prime, 8*k^4 + 1 = 253125001 is prime so 253125001 is the first entry.

%t Select[648 * Range[1000]^4 + 1, And @@ PrimeQ[# - {0, 2}] &] (* _Amiram Eldar_, Dec 26 2019 *)

%o (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(b","); ); ); print(); print(c) }

%Y Cf. A001097, A001359, A006512.

%K nonn

%O 1,1

%A _Cino Hilliard_, Jul 31 2006

%E Offset corrected and more terms added by _Amiram Eldar_, Dec 26 2019

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Last modified March 29 07:27 EDT 2024. Contains 371265 sequences. (Running on oeis4.)