HMMT 二月 2005 · TEAM2 赛 · 第 15 题

HMMT February 2005 — TEAM2 Round — Problem 15

[40] Suppose that S tiles the set of odd prime numbers. Prove that S has only one element. 2

解析

[40] Suppose that S tiles the set of odd prime numbers. Prove that S has only one element. Solution: Consider the set S equivalent to S that contains 3. If it contains 5 but 0 not 7, then the set S equivalent to S containing 7 must contain 9, which is not prime. 1 Likewise, S cannot contain 7 but not 5, because then the set S containing 5 must 0 1 contain 9. Suppose S contains 3, 5, and 7. Then any other set S of the tiling contains 0 1 elements p , p + 2, and p + 4. But not all of these can be prime, because one of them is divisible by 3. Finally, suppose S contains 3 and has second-smallest element p > 7. 0 Then the set S containing 5 does not contain 7 but does contain p + 2, and the set S 1 2 containing 7 contains p + 4. But as before, not all of p , p + 2, and p + 4 can be prime. Therefore S has no second-smallest element, so it has only one element. 4