| 發布者 | 內容列 |
aa963854116
Home away from home
註冊日: 2006-01-06
發表數: 392
從未發現之89號星座
|  一個競賽常見的題目類型 |  | a^2+2b^2+3c^2=10d^2
求證: 僅有一整數解a=b=c=d=0
是否能可使用無限遞降法完成?? |
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| 2007-02-21 16:35 |  |
isomorphism
Not too shy to talk
註冊日: 2007-01-30
發表數: 40
| | Re: 一個競賽常見的題目類型 | | What is 無限遞降法 in English? Infinite descent? |
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| 2007-02-22 09:57 | |
aa963854116
Home away from home
註冊日: 2006-01-06
發表數: 392
從未發現之89號星座
| | Re: 一個競賽常見的題目類型 | | 應該沒錯吧~ 這樣翻應該對...
或者, 可否用同餘解此題??
甚至有沒有其他方法? |
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| 2007-02-24 15:13 | |
jerry811215
Quite a regular
註冊日: 2006-12-30
發表數: 63
Chia-yi
| | Re: 一個競賽常見的題目類型 | | 那是什麼意思啊??可否解釋一下?? |
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| 2007-02-24 17:20 | |
isomorphism
Not too shy to talk
註冊日: 2007-01-30
發表數: 40
| | Re: 一個競賽常見的題目類型 | | I'll tackle this problem when I have more time. However, you can probably play around with mods, and show that you can get a smaller integer solution (usually by showing evenness and dividing by 2). Because you cannot have an infinite decreasing sequence of positive integers, you have reached a contradiction. |
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| 2007-02-26 19:25 | |
isomorphism
Not too shy to talk
註冊日: 2007-01-30
發表數: 40
| | Re: 一個競賽常見的題目類型 | | As a note, usually the concept of a "smaller integer solution set" is the most difficult aspect of a descent proof. You usually have to define that yourself.
For example, it may be (a',b')=(a/2,b/2). |
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| 2007-02-26 19:27 | |
abc831128
Just can't stay away
註冊日: 2007-06-26
發表數: 104
| | Re: 一個競賽常見的題目類型 | | mod 8之後
10dd-2bb==0,2 or 6
aa+3cc==0,1,3,4 or 7
a^2+2b^2+3c^2=10d^2 整數解
10dd-2bb==0 mod 8 ==> b,d都是偶數
aa+3cc==0 mod 8 ==> a,c都是偶數
==> 四數皆偶數
可使用無限遞降法完成
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| 2008-01-21 00:18 | |
