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

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?

137/2007=連分數{14,1,1,1,5,1,6}
137/2008=連分數{14,1,1,1,10,1,3}

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}

[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.

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]

{0,14,1,1,1,6} = 0+1/(14+(1+1/(1+1/(1+1/6))))) = 20/293

b. 待續...
#ed_op#P#ed_cl#b.#ed_op#BR#ed_cl##ed_op#BR#ed_cl#{14,1,1,1,6}=20/293#ed_op#BR#ed_cl#{14,1,1,1}=3/44#ed_op#BR#ed_cl##ed_op#BR#ed_cl#q=44(m-1)+293n#ed_op#BR#ed_cl#when n=4k-3, m=1 to 19k-14#ed_op#BR#ed_cl#when n=4k-2, m=1 to 19k-9#ed_op#BR#ed_cl#when n=4k-1, m=1 to 19k-4#ed_op#BR#ed_cl#when n=4k, m=1 to 19k#ed_op#/P#ed_cl##ed_op#P#ed_cl#gcd(m-1,n) should be 1#ed_op#/P#ed_cl##ed_op#P#ed_cl#p.s.1. need to be modified. The final answer should be 13937.#ed_op#BR#ed_cl#p.s.2. &nbsp;I don't know why#ed_op#/P#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl#

### [數學] c and d

#ed_op#DIV#ed_cl#c. #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#20+137=157#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#d. #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#20+137*(2007-1)=274842 is not a prime.#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#{14,1,1,1,7}= 23/337#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#23+137*(2007-1)=274845 is not a prime.#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#{14,1,1,1,8}= 26/381#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#DIV#ed_cl#26+137*(2007-1)=274848 is not a prime.#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#DIV#ed_cl#{14,1,1,1,9}= 29/337#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#29+137*(2007-1)=274851 is not a prime.#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#DIV#ed_cl#{14,1,1,1,10}= 32/337#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#32+137*(2007-1)=274854 is not a prime.#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#DIV#ed_cl#{14,1,1,1,6,2}= 43/630#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#43+137*(2007-1)=274865 is not a prime.#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;#ed_op#/DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#DIV#ed_cl#{14,1,1,1,7,2}= 49/718#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#p=49+137*(2007-1)=274871&nbsp;#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#q=[p*2007/137]+1 to [p*2008/137]+1#ed_op#/DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#=4026760 to 4028766#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#every (p,q)=1#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#and there are 2007 different values of q.#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#so, the smallest p=274871#ed_op#/DIV#ed_cl#

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

[quote=#ed_op#A href="mailto:G@ry"#ed_cl#G@ry#ed_op#/A#ed_cl#]#ed_op#BR#ed_cl#a. least possible q =&gt; least possible n#ed_op#BR#ed_cl#n≠1 [no value for 2008&gt;q&gt;2007]&nbsp; =&gt;&nbsp; n=2&nbsp; =&gt; 4016&gt;q&gt;4014#ed_op#BR#ed_cl#least possible q = 4015#ed_op#BR#ed_cl##ed_op#BR#ed_cl#check1: ∀ prime n, as 2007n &lt; q &lt; 2008n, q is relatively prime to n.#ed_op#BR#ed_cl#check2: 4015 is not divisible by 137, i.e. relatively prime to 137.#ed_op#BR#ed_cl#i.e. 4015 = q is prime to p=137x2=274.#ed_op#BR#ed_cl#[/quote]#ed_op#DIV#ed_cl#&nbsp;#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#how about p/q=20/293 ?#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#137/2007=連分數{14,1,1,1,5,1,6}#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#137/2008=連分數{14,1,1,1,10,1,3}#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#取連分數{14,1,1,1,6} =20/293#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#p.s.: I don't know why, but I ever&nbsp;saw this method in 數論淺談(written by 趙文敏).#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl#

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

QC 寫到:#ed_op#br#ed_cl#7. Suppose that p/q is a rational number such that 137/ 2008 &lt; p/q &lt; 137/2007, and that p and q are relatively prime positive integers.#ed_op#br#ed_cl#a. What is the least possible value for q?#ed_op#br#ed_cl#b. What is the 2007th smallest possible value for q?#ed_op#br#ed_cl#c. What is the smallest value for p for which there are at least two different possible values for q?#ed_op#br#ed_cl#d. What is the smallest value for p for which there are precisely 2007 different possible values for q?#ed_op#br#ed_cl#
#ed_op#br#ed_cl#137 is a prime number , #ed_op#span style="font-weight: bold; color: rgb(255, 0, 0);"#ed_cl#p is a multiple of 137#ed_op#/span#ed_cl#&nbsp; =&gt; q cannot be a multiple of 137#ed_op#br#ed_cl#Sorry, the above assumption is wrong and gave a wrong interpretation... : (#ed_op#br#ed_cl#

### [數學]FAMAT fall 2007 interschool (B)

#ed_op#DIV#ed_cl#7. Suppose that p/q is a rational number such that 137/ 2008 &lt; p/q &lt; 137/2007#ed_op#BR#ed_cl#, and that p and q are relatively prime positive integers.#ed_op#BR#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#a. What is the least possible value for q?#ed_op#BR#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#b. What is the 2007th smallest possible value for q?#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#c. What is the smallest value for p for which there are at least two different possible#ed_op#BR#ed_cl#values for q?#ed_op#BR#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#d. What is the smallest value for p for which there are precisely 2007 different possible#ed_op#BR#ed_cl#values for q?#ed_op#/DIV#ed_cl#