Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 5 down vote favorite
1

For any positive integer $n$, $i,j,k$ are also positive integers, and $0 <i,j,k < n$. How many solutions of the form $(i,j,k)$ are there for the equation $ijk = (n-i)(n-j)(n-k)$?

share|improve this question
2  
Let's get some trivial solutions out of the way. $n=2m$, $i=m-r$, $j=m$, $k=m+r$. Do you have a large supply of other solutions? – Gerry Myerson Jun 2 '12 at 12:43
2  
Here's another class: $y-1=(x-1)(x-2)/2$ then $n=xy,i=y,j=2y,k=n-x$, e.g. $(n,i,j,k)=(35,7,14,30)$. More generally if $y-1=(x-a)(x-b)/ab$ then $(n,i,j,k)=(xy,ay,by,xy-x)$ is a solution. e.g. $a=2,b=3,x=18,y=41$ and $(n,i,j,k)=(738,82,123,720)$. Also if $(n,i,j,k)$ then $(cn,ci,cj,ck)$ for positive integer $c$. – Zander Jun 2 '12 at 13:57

Know someone who can answer? Share a link to this question via email, Google+, Twitter, or Facebook.

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.