If $n>3$ is not prime, show that we can find positive integers $a,b,c$ such that $n=ab+bc+ca+1$.

Question

If $n>3$ is not prime, show that we can find positive integers $a,b,c$ such that $n=ab+bc+ca+1$.

When $n=4$, pick $a=b=c=1$. When $n=6,$ pick $a=2$ and $b=c=1$. When $n=8$, pick $a=b=c=2$. I don't get the pattern here. I know that if $n>3$ is not prime, then $n=xy,$ where $1<x<n$ and $1<y<n$. But how do I proceed from here?

score 4 · Accepted Answer · answered Jan 01 '20 at 23:28

4

Notice that $xy = (x-1)(y-1)+(y-1) + (x-1)+1.$ Choose $a=x-1,b=y-1, $ and $c=1$. All are positive as $x,y>1$.

answered Jan 01 '20 at 23:28

Simon Fraser

2,528

2

Nice and clean! – Wojowu Jan 01 '20 at 23:31
@Wojowu Special case $,p=1,$ of solutions in this answer in the dupe. See that thread and its links for much more. – Bill Dubuque Jan 02 '20 at 01:02

If $n>3$ is not prime, show that we can find positive integers $a,b,c$ such that $n=ab+bc+ca+1$.

1 Answers1