Forgot password
 Register account
View 1694|Reply 3

[数论] 解不定方程a,b,c为整数

[Copy link]

412

Threads

1432

Posts

3

Reputation

Show all posts

realnumber posted 2021-3-10 08:17 |Read mode
$a,b,c\in Z,\frac{b+c}{a}+\frac{a+c}{b}+\frac{a+b}{c}=1$,求a,b,c

461

Threads

957

Posts

4

Reputation

Show all posts

青青子衿 posted 2021-3-10 10:54
Last edited by 青青子衿 2021-3-10 11:02回复 1# realnumber
似乎是无整数解的
\begin{align*}
(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) &=n\\
\frac{a+b}{c}+\frac{b+c}{a}+\frac{a+c}{b} &= N =n-3
\end{align*}
Melvyn J. Knight's problem
asahi-net.or.jp/~KC2H-MSM/mathland/math03/xyz00.htm

A085514   Integers n representable as the product of the sum of three nonzero integers
with the sum of their reciprocals: n=(x+y+z)*(1/x+1/y+1/z).

根据OEIS上的数据,n=1, 9, 10, 11, 14, 15, 18, 26, 29, 30, ...
由于n取不到4,那么N取不到1。
oeis.org/A085514

412

Threads

1432

Posts

3

Reputation

Show all posts

original poster realnumber posted 2021-3-10 16:11
谢谢,看来被那2个初中,坑了

461

Threads

957

Posts

4

Reputation

Show all posts

青青子衿 posted 2021-3-10 18:00
Last edited by 青青子衿 2021-3-10 19:37
谢谢,看来被那2个初中,坑了
realnumber 发表于 2021-3-10 16:11
什么叫“二个初中”?

Quick Reply

Advanced Mode
B Color Image Link Quote Code Smilies
You have to log in before you can reply Login | Register account

$\LaTeX$ formula tutorial

Mobile version

2025-7-15 15:09 GMT+8

Powered by Discuz!

Processed in 0.011371 seconds, 24 queries