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??
高中

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isomorphism

2007-02-26 19:25
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.

isomorphism

2007-02-26 19:27
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).

abc831128

2008-01-21 00:18
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