Four tuple/Modulus of differences/Introduction/Exercise

We consider the mapping

$\Psi \colon \mathbb {N} ^{4}\longrightarrow \mathbb {N} ^{4}$

that assigns to a four tuple ${}(a,b,c,d)$ the four-tuple

$(\vert {b-a}\vert ,\vert {c-b}\vert ,\vert {d-c}\vert ,\vert {a-d}\vert ).$

We denote by ${}\Psi ^{n}$ the ${}n$-th fold composition of ${}\Psi$ with itself.

1. Compute

   $\Psi (6,5,2,8),\,\Psi ^{2}(6,5,2,8),\,\Psi ^{3}(6,5,2,8),\,\Psi ^{4}(6,5,2,8)\,...$

   until the result is the zero-tuple ${}(0,0,0,0)$.

2. Compute

   $\Psi (1,10,100,1000),\,\Psi ^{2}(1,10,100,1000),\,\Psi ^{3}(1,10,100,1000),\,\Psi ^{4}(1,10,100,1000)\,...$

   until the result is the zero-tuple ${}(0,0,0,0)$.

3. Show that ${}\Psi ^{4}(0,0,n,0)=(0,0,0,0)$ for every ${}n\in \mathbb {N}$.

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