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---- · 由 **----** 於星期三三月 19, 2003 11:11 pm

k是一個大於13的整數，p_k 是比k小的數中最大的質數。若n是一合成數，證明：
(a) 若n=2(p_k), 則n不整除(n-k)!
(b) 若n > 2(p_k)，則n整除(n-k)!

scsnake · 由 **scsnake** 於星期四四月 03, 2003 9:28 pm

這是APMO2003考題，(a)我會，但(b)我只會證一半^^

---- · 由 **----** 於星期四四月 03, 2003 9:33 pm

我也會(a)，但不會(b)...^^|||
(a)我用wilson, 你呢？

scsnake · 由 **scsnake** 於星期四四月 03, 2003 9:48 pm

我(a)好像是列表找順序，再說明，不過回想起來好像有wilson的味道～

---- · 由 **----** 於星期四四月 03, 2003 10:03 pm

my method~
2|(n-k)!
k+ pk >n
pk > n-k
pk-1 >= n-k
by wilson theorem
(pk-1)! congruent to -1 (mod pk)
as n-k <= pk-1, so
n-k will never congruent to 0 (mod pk)

scsnake · 由 **scsnake** 於星期四四月 03, 2003 10:44 pm

呼，比我的精簡多了，你可以自己加自己5000元^^

---- · 由 **----** 於星期四四月 03, 2003 10:46 pm

...
但b part 想不出來~~

E.T · 由 **E.T** 於星期四四月 03, 2003 11:11 pm

Sorry ar

I have some question .

Q1. What is wilson theorem ?

Q2. pk > n-k
      pk-1 >= n-k     ???
      This step ???

Q3. xxxx ( mod xx)    ???
      What is mod ?

Raceleader · 由 **Raceleader** 於星期四四月 03, 2003 11:20 pm

chung 5000

---- · 由 **----** 於星期四四月 03, 2003 11:25 pm

Q1. What is wilson theorem ?

If p is a prime, we have
(p-1)!+1=0(mod p)

Q2. pk > n-k
pk-1 >= n-k ???
This step ???
Because pk, n, k are integers, when pk>n-k, pk-1>=n-k

Q3. xxxx ( mod xx) ???
What is mod ?

We say
a congruent to b (mod c) when
the remainder of a divided by c equals to the remainder of b divided by c.
It's similar to that of pascal.

You can express by this way:
a-b=ck (a,b,c,k are integers)
then a=b(modc)

Raceleader · 由 **Raceleader** 於星期四四月 03, 2003 11:33 pm

E.T:
In School syllabus, we do not need to study mod.
You can learn this by reading books. This is difficult to tell all information of mod here.

In Chinese, mod is 同餘

stelny · 由 **stelny** 於星期五四月 04, 2003 12:48 am

I am stupid, don't know how to calculate.

--- · 由 **---** 於星期日四月 06, 2003 3:22 pm

好像還有一個條件: p_k > 3k/4

加上這個應該可以解吧.

Raceleader · 由 **Raceleader** 於星期日四月 06, 2003 3:23 pm

This question is copied from APMO

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[數學]整除, 10000元

[數學]整除, 10000元