A159625 Numbers n such that 2^x + 3^y is never prime when max(x,y) = n.

1679, 1743, 4980, 4982, 5314, 5513, 5695, 6100, 6578, 7251, 7406, 7642, 8218, 8331, 9475, 9578, 9749, 10735

Offset: 1

Author

David Broadhurst, Apr 17 2009

Keywords

• nonn
• more
• hard

Comments

Mark Underwood found that for each nonnegative integer n < 1421 there is at least one prime of the form 2^m + 3^n or 2^n + 3^m with m not exceeding n.

This sequence consists of numbers for which there is no such prime.

David Broadhurst estimated that a fraction in excess of 1/800 of the natural numbers belongs to this sequence and found 17 instances with n < 10^4.

For each of the remaining 9983 nonnegative integers n < 10^4, a prime or probable prime of the form 2^x + 3^y was found with max(x,y) = n.

Each probable prime was subjected to a combination of strong Fermat and strong Lucas tests.

Examples

			a(3) = 4980, since there is no prime of the form 2^m + 3^4980 or 2^4980 + 3^m with m < 4981 and 4980 is the third number n such that 2^x + 3^y is never prime when max(x,y) = n.
		

Links

• Broadhurst's heuristic in the PrimeNumbers list. [Broken link]
• Maximilian Hasler, Mike Oakes, Mark Underwood, David Broadhurst and others, Primes of the form (x+1)^p-x^p, digest of 22 messages in primenumbers Yahoo group, Apr 5 - May 7, 2009. [Cached copy]
• Underwood's posting in the PrimeNumbers list
• A list of 9983 primes or probable primes for the excluded cases with n < 10^4

Crossrefs

Cf. A159270, A159266, A123359.

Extensions

a(18) from Giovanni Resta, Apr 09 2014