Re: [問題] Number Theory

[myflame (是否繼續墮落)]

※ 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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推 Dawsen:obvious那句是why
推 LPH66:當n夠大時就不夠obvious了
→ LPH66:可能存在某個k<n使a|a+kb
推 myflame:謝謝。假解法 哈哈
推 myflame:不過我覺得有機會修正，我再想想歐。