WDS quoting T.Tao: N = 47867742232066880047611079 mod (2*3*5*7*11*13*17*19*31*37*41*61*73*97*109*151*241*257*231) Dan Asimov: According to my calculations, that last factor should be 331, not 231. --Tao's paper says 231, but, since 3|231, it is not prime, while Tao claims it is prime. Oops. And this correction does indeed de-mystify the connection to Zhi-Wei Sun's earlier work. And then, Sun's claim is even stronger than Tao's claim. Sun says every such N has the property that if you add or subtract any prime power from it (note: not just powers of 2, but of any prime) then you get a number whose absolute value is composite. The way to prove this in Tao's power-2 case is, you show some set of powers of 2 is going to lead to divisibility by 331, some other set by 257, etc, and show the union of all these sets is everything. This suggests this question: are there an infinite set (or finite nonempty set) of odd N such that if you add or subtract any power>1 of any integer, you always get a composite? I have an embarrassingly simple proof that both answers are NO... but if we demand the power be cubic or higher, then I still think the answers are NO, but now I have no proof. Sun also conjectures that every odd N>8 is the sum of a prime and three positive powers of 2. But (Crocker 1971 proved) that summing a prime plus two positive powers of 2 does not suffice to represent an infinite set of odd N. Sun also conjectures that every odd N>4 is the sum of a prime and two positive Fibonacci numbers. Has been verified for 4<N<10^14. He offers a $5000 prize for settling this.