PUMaC 2017 · 团队赛 · 第 1 题

PUMaC 2017 — Team Round — Problem 1

(3) Call an ordered triple ( a, b, c ) of integers feral if b − a , c − a and c − b are all prime. Find the number of feral triples where 1 ≤ a < b < c ≤ 20. 100 ∑

解析

It’s essential to note that the pairs of twin primes in our range are (3 , 5) , (5 , 7) , (11 , 13) , (17 , 19). The additional prime therein is 2. If p , p are primes with | p − p | prime, then the correct ordering of ( k, k ± p , k ± p ) is feral. 1 2 1 2 1 2 If p + p is prime then the correct ordering of ( k, k ± 2 , k ∓ p ) is feral. The former condition 1 2 1 is satisfied by the twin primes; the latter by 2 and the smaller in each pair of twin primes. ( k, k + 3 , k + 5) , 1 ≤ k ≤ 15 → 15 ( k, k − 3 , k − 5) , 6 ≤ k ≤ 20 → 15 ( k, k + 5 , k + 7) , 1 ≤ k ≤ 13 → 13 ( k, k − 5 , k − 7) , 8 ≤ k ≤ 20 → 13 ( k, k + 11 , k + 13) , 1 ≤ k ≤ 7 → 7 ( k, k − 11 , k − 13) , 14 ≤ k ≤ 20 → 7 ( k, k + 17 , k + 19) , 1 ≤ k ≤ 1 → 1 ( k, k − 17 , k − 19) , 20 ≤ k ≤ 20 → 1 This gives a total of 2(15 + 13 + 7 + 1) = 72 triples. We can see that there are no other solutions, since these are given by letting p = b − a , 1 p = c − a and a = k , and that there is no double-counting. 2 Problem written by Zack Stier 100 100 100