HMMT 十一月 2015 · 冲刺赛 · 第 35 题

HMMT November 2015 — Guts Round — Problem 35

[ 20 ] Let n be the smallest positive integer such that any positive integer can be expressed as the sum 1 a of n integer 2015th powers. Find n . If your answer is a , your score will be max(20 − | log | , 0), 10 5 n rounded up.

解析

[ 20 ] Let n be the smallest positive integer such that any positive integer can be expressed as the sum 1 a of n integer 2015th powers. Find n . If your answer is a , your score will be max(20 − | log | , 0), 10 5 n rounded up. Proposed by: Alexander Katz ⌊ ⌋ ( ) 2015 3 2015 Answer: 2 + − 2 2 ⌊ ⌋ ( ) k 3 k In general, if k ≤ 471600000, then any integer can be expressed as the sum of 2 + − 2 integer 2 k th powers. This bound is optimal. The problem asking for the minimum number of k -th powers needed to add to any positive integer is called Waring’s problem.