HMMT 十一月 2015 · 冲刺赛 · 第 35 题
HMMT November 2015 — Guts Round — Problem 35
题目详情
英文原题
- [ 20 ] Let n be the smallest positive integer such that any positive integer can be expressed as the sum
1 aof n integer 2015 th powers. Find n . If your answer is a , your score will be max(20 − | log | , 0),
5 n 10
rounded up.
解析
英文解析
- [ 20 ] Let n be the smallest positive integer such that any positive integer can be expressed as the sum
1 aof n integer 2015 th powers. Find n . If your answer is a , your score will be max(20 − | log | , 0),
5 n 10
rounded up.
Proposed by: Alexander Katz
⌊ ⌋
( )
2015
20153
Answer: 2 + − 2
⌊ ⌋2
( )
3 kk
In general, if k ≤ 471600000, then any integer can be expressed as the sum of 2 + − 2 integerk th powers. This bound is optimal.2
The problem asking for the minimum number of k -th powers needed to add to any positive integer iscalled Waring’s problem.