I am trying to solve the following problem.
If $\psi(s) = \frac{s(s-1)}2$. I write $f(s,k) = (\psi(s),\psi(s-k))$, where $k$ is a fixed positive integer. Let $K$ be the image of $f_{s,k}$. If $s>3$ and $k>1$
- Show that $f_{s,k}$ is periodic and in particular $K$ is finite. This is the same as showing that $f_{s+t,k} = f_{s,k}$ for some $t$.
- Show that $K$ has a greatest element $g$ and a least element $1$
- Show that $K$ is the divisor lattice of $g$.
- Is it true that $3|g$ ?
Here are some initial values:
- $f_{s,2} = 1,1,3,1,1,3,1,1,3,1,1,3,....$ so $K=[1,3]$
- $f_{s,3} = 1,3,3,2,3,3,1,6,3,1,3,6,....$ so $K=[1,2,3,6]$
- $f_{s,4}$ and $f_{s,5 }$ we have $K=[1,2,3,5,6,10,15,30]$
- $f_{s,6}$ we have $K=[1,2,3,5,7,15,21,35,105]$
- $f_{s,7}$ we have $K=[1,2,3,4,6,7,14,21,28,42,84]$
Also I am assuming "periodic" is the correct term here used to describe $f_{s,k}$