  The usual product $m\cdot n$ on $\mathbb{Z}$ can be viewed as the sum of $n$
terms of an arithmetic progression whose first term is $a_{1}=m-n+1$ and whose
difference is $d=2$. Generalizing this idea, we define new similar product
mappings, and we consider new arithmetics that enable us to extend
Furstenberg's theorem of the infinitude of primes. We also review the classic
conjectures in the new arithmetics. Finally, we make important extensions of
the main idea. We see that given any integer sequence, the approach generates
an arithmetic on integers and a similar formula to $\mathbb{Z} \setminus
\{-1,1\}$ arises.
