PUMaC 2012 · 数论（A 组） · 第 2 题
PUMaC 2012 — Number Theory (Division A) — Problem 2
题目详情
- [ 3 ] How many ways can 2 be expressed as the sum of four (not necessarily distinct) positive squares? x x k k
解析
2 . 012 × 10 ≤ x ≤ 2 . 013 × 10 Because all values are greater than or equal to 1, we can take the logarithm of each part of the inequality, yielding s + log 2 . 012 ≤ n log x ≤ s + log 2 . 013 . 10 10 10 If log x is irrational, then we are guarenteed to find a n to satisfy these conditions, by the 10 Equidistribution Theorem (this is also intuitively obvious). The only integers x for which 2 log x is rational are powers of 10. For x = 1 , 10 , 100 , 1000, we can see that the leading digit 10 is always 1, which means x is not a leader for these four numbers. All other numbers have an irrational common log, so there are 2012 − 4 = 2008 leaders in the set. Problem contributed by Wesley Cao. p n − 1