证明每一个大於32的整数都可分解成若干自然数的和，这些自然数的倒数和是1

野菜花 · 发表于 2005-12-19 22:03:38

如果正整数 n 可以表示成

n=a1+a2+...+akwww.ddhw.com

其中 a1, a2, ...,ak 是满足

1/a1+1/a2+...+1/ak=1

的自然数(不一定相异)，那么我们称 n 是好数。

已知整数33 至 73 是好数，证明每个大於32的整数都是好数。

IQ75 · 发表于 2005-12-19 22:32:38

brief

1/2 + ...

1/3 + 1/6 + ...

QFT · 发表于 2005-12-20 05:44:45

it takes me a while to show that 33 至 73 是好数.

野菜花 · 发表于 2005-12-20 08:04:05

all integers beteen 33 to 73 are good numbers.

Have you proved all n>32 are good numbers?

我要说 · 发表于 2005-12-20 08:18:54

but he couldn't be bothered to write down the details of his proof. I don't want to fill out the details for other people either, so I will just give a bit more hints than IQ75

:

Inductively, for even number N = (N-2) + 2, apply decomposition to (N-2)/2 and add that to 1/2.
For odd number N = (N-9) + 3 + 6, apply decomposition to (N-9)/2 and add that to 1/3 + 1/6. Hope this suffices.

QFT · 发表于 2005-12-20 08:20:15

following IQ75, we can show that all > 32 are good numbers.
if n is a good number, then 2n + 8 and 2n + 9 are both good numbers.

QFT · 发表于 2005-12-20 08:21:59

Just notice your post.

husonghu · 发表于 2005-12-20 08:29:36

看来本坛的数学高手是越来越多了

野菜花 · 发表于 2005-12-20 08:58:48

Smart point!

野菜花 · 发表于 2005-12-20 09:00:06

Great!

野菜花 · 发表于 2005-12-20 09:08:19

想过的也会心有灵犀。可是对本来不懂的人就又是一题脑筋一转啦！

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证明每一个大於32的整数都可分解成若干自然数的和，这些自然数的倒数和是1

回复：证明每一个大於32的整数都可分解成若干自然数的和，这些自然数的倒数和是1

it takes me a while to show that 33 至 73 是好数.

This is given [;)] It is not easy to verify that

I think IQ75 had already got an answer,

回复：This is given [;)] It is not easy to verify tha

Just notice your post.[@};-][>:D<]

看来本坛的数学高手是越来越多了[@};-][@};-][@};-][@};-]

Smart point! [@};-]

Great! [@};-][@};-]

哈哈，都很厉害！[@};-][@};-]都有大师风范，写一两行关键，知道答案的自然能看懂，