Re: [代數]請前輩指點一下整數証明～急！謝謝
※ 引述《rfvbgtsport (uygh)》之銘言：
: 若a，b，c為整數，且a／b＋b／c＋c／a為整數，b／a
: ＋c／b＋a/c亦為整數，証明丨a丨＝丨b丨＝丨c丨
: 請高手幫忙一下，謝謝
Another proof:
Suppose otherwise.
By permuting a,b,c, we may assume there is a prime p such that
v_p(a) >= v_p(b) >= v_p(c) (*)
and not both equality in (*) --- (**).
[Recall: v_p(a)=sup{n>=0 : p^n divides a} is the p-adic valuation on Z,
which can be extended to Q by v_p(m/n)=v_p(m)-v_p(n)]
Then from
v_p(a/b+b/c+c/a) >= 0, v_p(c/a)<0, v_p(a/b)>=0
we have: 0 > v_p(b/c) = v_p(c/a) from the ultrametric property
i.e.: v_p(b)-v_p(c) = v_p(c)-v_p(a)
i.e.: 2*v_p(c) = v_p(a)+v_p(b) (***)
(*) and (***) gives v_p(a), v_p(b), v_p(c) all equal, contradicting (**).
So |a|=|b|=|c|.
--
『我思故我在』怎樣從法文變成拉丁文的：
je pense, donc je suis --- René Descartes, Discours de la Méthode (1637)
ego sum, ego existo --- ____, Meditationes de Prima Philosophia (1641)
ego cogito, ergo sum --- ____, Principia Philosophiae (1644)
--
※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 140.112.101.8
※ 文章網址: https://www.ptt.cc/bbs/Math/M.1475645024.A.6F9.html
討論串 (同標題文章)
本文引述了以下文章的的內容：
完整討論串 (本文為第 3 之 3 篇)：