[數學]FAMAT fall 2007 interschool (B)

[數學]FAMAT fall 2007 interschool (B)

QC 於 星期五 十一月 02, 2007 10:34 pm


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?
CCC

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Re: [數學]FAMAT fall 2007 interschool (B)

G@ry 於 星期日 十一月 04, 2007 2:40 am


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?

137 is a prime number , p is a multiple of 137  => q cannot be a multiple of 137
Sorry, the above assumption is wrong and gave a wrong interpretation... : (
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Re: [數學]FAMAT fall 2007 interschool (B)

QC 於 星期日 十一月 04, 2007 11:06 am


[quote=G@ry]
a. least possible q => least possible n
n≠1 [no value for 2008>q>2007]  =>  n=2  => 4016>q>4014
least possible q = 4015

check1: ∀ prime n, as 2007n < q < 2008n, q is relatively prime to n.
check2: 4015 is not divisible by 137, i.e. relatively prime to 137.
i.e. 4015 = q is prime to p=137x2=274.
[/quote]
 
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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[數學] c and d

QC 於 星期日 十一月 04, 2007 1:58 pm


c.
20+137=157
 
d.
20+137*(2007-1)=274842 is not a prime.
{14,1,1,1,7}= 23/337
23+137*(2007-1)=274845 is not a prime.
{14,1,1,1,8}= 26/381
26+137*(2007-1)=274848 is not a prime.
{14,1,1,1,9}= 29/337
29+137*(2007-1)=274851 is not a prime.
{14,1,1,1,10}= 32/337
32+137*(2007-1)=274854 is not a prime.
{14,1,1,1,6,2}= 43/630
43+137*(2007-1)=274865 is not a prime.
 
{14,1,1,1,7,2}= 49/718
p=49+137*(2007-1)=274871 
q=[p*2007/137]+1 to [p*2008/137]+1
=4026760 to 4028766
every (p,q)=1
and there are 2007 different values of q.
 
so, the smallest p=274871
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QC 於 星期日 十一月 04, 2007 3:32 pm


b.

{14,1,1,1,6}=20/293
{14,1,1,1}=3/44

q=44(m-1)+293n
when n=4k-3, m=1 to 19k-14
when n=4k-2, m=1 to 19k-9
when n=4k-1, m=1 to 19k-4
when n=4k, m=1 to 19k

gcd(m-1,n) should be 1

p.s.1. need to be modified. The final answer should be 13937.
p.s.2.  I don't know why

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G@ry 於 星期一 十一月 05, 2007 2:10 am


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Re: [數學]FAMAT fall 2007 interschool (B)

G@ry 於 星期四 十二月 20, 2007 1:13 am


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).

--------------------------------------
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...
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