This is A137985 in the OEIS, where you could have found the Tao reference Feel free to your comments there Neil On Fri, Mar 29, 2013 at 12:53 PM, Warren D Smith <warren.wds@gmail.com>wrote:
Each of the following is a prime number P, such that if any single bit of P's binary representation is changed, then you no longer have a prime. (And this is the full list of such P<10000.)
127, 173, 191, 223, 233, 239, 251, 257, 277, 337, 349, 373, 431, 443, 491, 509, 557, 653, 683, 701, 733, 761, 787, 853, 877, 1019, 1193, 1201, 1259, 1381, 1451, 1453, 1553, 1597, 1709, 1753, 1759, 1777, 1973, 2027, 2063, 2333, 2371, 2447, 2633, 2879, 2917, 2999, 3083, 3181, 3209, 3313, 3511, 3593, 3643, 3767, 3779, 3851, 3877, 3889, 3967, 4013, 4177, 4283, 4441, 4451, 4561, 4597, 4603, 4679, 4813, 4889, 4951, 5051, 5099, 5209, 5323, 5557, 5801, 5867, 6007, 6073, 6151, 6203, 6211, 6287, 6323, 6379, 6481, 6521, 6971, 6977, 6997, 7027, 7039, 7043, 7103, 7109, 7151, 7207, 7297, 7307, 7331, 7369, 7507, 7573, 7583, 7841, 7883, 8017, 8087, 8111, 8171, 8231, 8243, 8311, 8363, 8627, 8747, 8831, 8849, 8867, 8923, 9137, 9151, 9161, 9319, 9323, 9697, 9767
The Mersenne primes P=2^p-1 also have this "every bit matters" property when p = 7, 31, 127, 607, 1279, 4423 for the p<5000.
I would presume that asymptotically, a constant fraction C of all primes are "every bit matters" primes. What is that constant? I would guess that C = exp( (-1/ln2) * (1-1/2)/(1-1/3) * (1-1/4)/(1-1/5) * (1-1/6)/(1-1/7) * ... ) = exp( (-1/ln2) * (1*3)/2^2 * (3*5)/4^2 * ... * (p-2)*p/(p-1)^2 * ... ) = exp( (-1/ln2) * 0.660162) = 0.38581 the product being over all odd primes p.
However, my guess seems to be wrong in view of the actual counts 89486 / 664579 = 0.13465 from the primes below 10^7 794760 / 5761455 = 0.13794 from the primes below 10^8
I think I see my error now... getting this right should be tricky.
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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