QC 寫到:
7. Suppose that p/q is a rational number such that 137/ 2008 < p/q < 137/2007, and that p and q are relatively prime positive integers.
a. What is the least possible value for q?
b. What is the 2007th smallest possible value for q?
c. What is the smallest value for p for which there are at least two different possible values for q?
d. What is the smallest value for p for which there are precisely 2007 different possible values for q?
how about p/q=20/293 ?
137/2007=連分數{14,1,1,1,5,1,6}
137/2008=連分數{14,1,1,1,10,1,3}
取連分數{14,1,1,1,6} =20/293
p.s.: I don't know why, but I ever saw this method in 數論淺談(written by 趙文敏).
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Well, I try to explain as possible.
First, it should be that
137/2007=連分數{0,14,1,1,1,5,1,6},
137/2008=連分數{0,14,1,1,1,10,1,3}
首先,連分數就是最漸近的分數的表示方法,由於此為連分數的基礎,而且解釋很長,若不明白請看前面的那個連結了解...
[前面的連結解釋得很好,而且小弟不想copy&paste, 故自己看看吧]
a.
由連分數的定義得知,連分數的單數位置對應分子,而雙數位置對應分母,故單數位置數字越大或雙數位置越小,則數字越大:
即 {a+1,b,c} > {a,b,c+1} > {a,b,c} > {a,b+1,c}...
而唯一的例外是 {... ,n,1} = {... ,n+1} -- *****
而數連分數的分子分母組成得知,若有同樣的前置數字,則後置數字的大小與對應的分子/分母數字的大小有直接關係:
即 {a,b,c,d,2}的分子比{a,b,c,d,3}的分子小
而若有同樣的前置數字,則有較多的後置數字會比較小的所對應的分子及分母大:
即 {a,b,c,d}的分子及分母
{0,14,1,1,1,10,1,3} < 範圍 <0>
1. 短 => {0,14,1,1,1,x} 為範圍內最短的連分數, 而 5 <x> x 為盡量小 => x=6
[註:{0,14,1,1,1,5,1,6} > {0,14,1,1,1,5,1} = {0,14,1,1,1,6} || *****
而且{0,14,1,1,1,5,1,6} 比 {0,14,1,1,1,5,1} = {0,14,1,1,1,6}長]
於範圍內最小分母(亦同時是最小分子,但不一定是唯一最小分子)的連分數表示為
{0,14,1,1,1,6} = 0+1/(14+(1+1/(1+1/(1+1/6))))) = 20/293
b. 待續...