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.

253125001, 10871635969, 14688294409, 168573727369, 196730062849, 248935680001, 528593507209, 759035205001, 956311308289, 1602486789769, 2451216826369, 9613393373449, 18132940558729, 60600405623689, 142671521205001, 178044790376449, 261461826945289, 290127048939649

Offset

1,1

Comments

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

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

Mathematica

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

Prog

(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) }

Crossrefs

Cf. A001097, A001359, A006512.

Sequence in context: A015393 A262989 A119859 * A210683 A271111 A204415

Adjacent sequences: A119857 A119858 A119859 * A119861 A119862 A119863

Keyword

nonn

Author

Cino Hilliard, Jul 31 2006

Extensions

Offset corrected and more terms added by Amiram Eldar, Dec 26 2019

Status

approved