A071781 Primes p with p-2^e and p+2^e prime for some exponent e.

5, 7, 11, 67, 32771

Offset: 1

Author

Rick L. Shepherd, Jun 05 2002

Keywords

• hard
• nonn

Comments

For each n, p-2^e,p,p+2^e is thus an arithmetic progression of primes with difference 2^e. Note that for each n=1,2,3,4,5, only one such e exists and p-2^e=3. There are no other terms up to 20000000.

For all terms, p-2^e must, in fact, be 3 (as one of p-2^e, p and p+2^e is divisible by 3). Each corresponding arithmetic progression of primes has length 3 (p+2^(e+1) is also divisible by 3). Any additional term is too large to include here. Equivalently, this sequence is primes of the form 3+2^e such that 3+2^(e+1) is also prime; i.e., 3+2^A057732(k) is a term iff A057732(k+1) = A057732(k) + 1. Thus much more efficient than the PARI program below is to extend A057732 and examine its terms. - Rick L. Shepherd, Jun 20 2008

Examples

			67 is a term because 67 is prime and there exists e=6 such that both 67-2^6=67-64=3 and 67+2^6=67+64=131 are primes. 32771 is a term because 32771 is prime and there exists e=15 such that both 32771-2^15=32771-32768=3 and 32771+2^15=32771+32768=65539 are primes. Thus 3,67,131 and 3,32771,65539 are two sequences of primes in arithmetic progression with differences 2^6 and 2^15, respectively.
		

Crossrefs

Cf. A056206, A056208.

Cf. A057732.

Programs

PARI

for(p=5,20000000,if(isprime(p),e=1; while(p-2^e>1,if(isprime(p-2^e)&&isprime(p+2^e),print1(p,","); break,e++))))