Keith F. Lynch kfl at KeithLynch.net "Fred W. Helenius" <fredh at ix.netcom.com> wrote: 271129 is such a number; it is prime and 271129 + 2^k is always divisible by (at least) one of 3, 5, 7, 13, 17 and 241.
Tao in his paper page 1 (he says this result is due to Sun, but it actually is not immediate from what Sun says, you must have to do some extra work) claims that every N with N = 47867742232066880047611079 mod (2*3*5*7*11*13*17*19*31*37*41*61*73*97*109*151*241*257*231) has the property that N+2^k and |N-2^k| are always composite for every k>=0, in other words for the prime such N, "every bit matters" including(!) leading 0s, so REALLY every bit matters (I originally had in mind not counting leading 0s but Tao permits us)... and something stronger also is true since you can change 1s to 0s and 0s to 1s, but also 1s to 2s and 0s to (-1)s, eh? Evidently this was inspired by Sierpinski, as Helenius points out... but Tao never mentions Sierpinski in his paper so it is a good thing Helenius told us this. (Tao actually proves his analogous results using radices>2 without using just covering congruences, he also uses sieving arguments.)