作者myflame (是否繼續墮落)
看板IMO_Taiwan
標題Re: [問題] Number Theory
時間Sat Feb 12 14:05:42 2005
※ 引述《darkseer (進入無限期公假)》之銘言：
: This problem is from AoPS.
: http://www.artofproblemsolving.com/Forum/viewtopic.php?t=26137
: n, a, and b are natural numbers. Prove that:
: 1 1 1
: --- + --- + ... + ---- isn't natural.
: a a+b a+nb
: By being natural, we mean that number is a positive integer.
: Your English may not be as poor as mine, but it takes me 5 mins to
: understand it orz...
if there's a pair of numbers (a.b) that makes the sum natural
let (a,b)=k and let a=xk, b=yk
1 1 1 1 1 1 1 1 1
--- + --- + ... + ---- = ---- + ----- + ... + ------ = --- (--- + --- + ... )
a a+b a+nb xk xk+yk xk+nyk k x x+y
1 1 1
so --- + --- + ... + ---- is also natural.
x x+y x+ny
that means we can let (a,b)=1 and find all answers.
but when (a,b)=1 a,a+b,a+2b,...a+nb are relatively prime
1 1 1 (a+b)(a+2b)...(a+nb) + a(...)
--- + --- + ... + ---- = -------------------------------
a a+b a+nb a(a+b)(a+2b)...(a+nb)
it's obvious that it's not an integer.
1 1 1
==> --- + --- + ... + ---- isn't natural for every (a,b,n)
a a+b a+nb
--
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◆ From: 218.164.122.21
1F：推 Dawsen:obvious那句是why 218.167.198.65 02/12
2F：推 LPH66:當n夠大時就不夠obvious了 61.62.178.179 02/12
3F：→ LPH66:可能存在某個k<n使a|a+kb 61.62.178.179 02/12
4F：推 myflame:謝謝。假解法 哈哈 218.164.122.21 02/12
5F：推 myflame:不過我覺得有機會修正，我再想想歐。 218.164.122.21 02/12