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

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

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

QC

### 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 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... : (
☆子 是也

G@ry

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

[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}

p.s.: I don't know why, but I ever saw this method in 數論淺談(written by 趙文敏).

CCC

QC

### [數學] c and d

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
CCC

QC

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

CCC

QC

☆子 是也

G@ry

### 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?

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}

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}

[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...
☆子 是也

G@ry