A curiosity: There seem to be 1.5 times as many primes of the form 2^N-3 as there are of the form 2^N+3. 2^N+3 is prime for N = 1 2 3 4 6 7 12 15 16 18 28 30 55 67 84 228 390 784 while 2^N-3 is prime for N = 3 4 5 6 9 10 12 14 20 22 24 29 94 116 122 150 174 213 221 233 266 336 452 545 689 694 850 (up to N=1000). The former sequence is divisible by 7 when N = 2 mod 3, while the latter is never a multiple of 7. Small primes other than 7 seem to strike both sequences equally. The large gaps (390-784, and 29-94) and general bunchiness (213, 221, 233; 689, 694) makes it hard to draw any firm conclusions. Rich
On 10/7/06, Schroeppel, Richard <rschroe@sandia.gov> wrote:
The former sequence is divisible by 7 when N = 2 mod 3, while the latter is never a multiple of 7. Small primes other than 7 seem to strike both sequences equally. The large gaps (390-784, and 29-94) and general bunchiness (213, 221, 233; 689, 694) makes it hard to draw any firm conclusions.
I always feel that except for "obvious" small divisors, primes act like they are random, so based on this observation about mod 3, it makes perfect sense that there'd be about 2/3 as many primes of this form as of the other. To me, that's what's so magical about primes: they are simultaneously entirely predictable and yet behave as though they were completely random: compare the actual primes, the "lucky" numbers, and the things you get if you use a probabilistic version of divisibility where, for example, every number > 2 has a random 1/2 chance of being "divisible" by 2. It turns out that this is self-correcting. If 3 turns out to be "composite" by this random method, then for a while there are more "primes" than expected, because that 1/3 of numbers aren't crossed out as you "sieve", but then eventually those extra "primes" hit more targets, and in the long run asymptotically you still get pretty much the same old prime number theorem. Integers really are amazing things. --Joshua Zucker
The Sequence database has more information: A057732 2^N+3 prime 1, 2, 3, 4, 6, 7, 12, 15, 16, 18, 28, 30, 55, 67, 84, 228, 390, 784, 1110, 1704, 2008, 2139, 2191, 2367, 2370, 4002, 4060, 4062, 4552, 5547, 8739, 17187, 17220, 17934, 20724, 22732, 25927, 31854, 33028, 35754, 38244, 39796, 40347, 55456, 58312, 122550 A050414 2^N-3 prime 3, 4, 5, 6, 9, 10, 12, 14, 20, 22, 24, 29, 94, 116, 122, 150, 174, 213, 221, 233, 266, 336, 452, 545, 689, 694, 850, 1736, 2321, 3237, 3954, 5630, 6756, 8770, 10572, 14114, 14400, 16460, 16680 2^N-3 is still leading the race at 17000, but the ratio has dropped quite a bit from 1.5. The bunchiness winner is with 2^N+3: N = both 4060 and 4062 give (probable) primes. Technical note: I've been using "probable prime" tests, and the notes in the sequence database mention this too. Rich -----Original Message----- From: Schroeppel, Richard Sent: Sat 10/7/2006 9:37 PM To: math-fun@mailman.xmission.com Cc: Schroeppel, Richard; rcs@cs.arizona.edu Subject: Primes in 2^N +-3 A curiosity: There seem to be 1.5 times as many primes of the form 2^N-3 as there are of the form 2^N+3. 2^N+3 is prime for N = 1 2 3 4 6 7 12 15 16 18 28 30 55 67 84 228 390 784 while 2^N-3 is prime for N = 3 4 5 6 9 10 12 14 20 22 24 29 94 116 122 150 174 213 221 233 266 336 452 545 689 694 850 (up to N=1000). The former sequence is divisible by 7 when N = 2 mod 3, while the latter is never a multiple of 7. Small primes other than 7 seem to strike both sequences equally. The large gaps (390-784, and 29-94) and general bunchiness (213, 221, 233; 689, 694) makes it hard to draw any firm conclusions. Rich
participants (2)
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Joshua Zucker -
Schroeppel, Richard