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

253124999, 10871635967, 14688294407, 168573727367, 196730062847, 248935679999, 528593507207, 759035204999, 956311308287, 1602486789767, 2451216826367, 9613393373447, 18132940558727, 60600405623687

Offset: 1

Author

Cino Hilliard, Jul 31 2006

Keywords

• nonn

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.

Examples

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

Links

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

Programs

Mathematica

Select[8*(3Range[600])^4,And@@PrimeQ[{#+1,#-1}]&]-1 (* Harvey P. Dale, Feb 14 2013 *)

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) }

Extensions

Offset corrected by Amiram Eldar, Dec 24 2019