An operation (or a relation) on the set $\NN$ of nonnegative
integers is called arithmetical if it is first order
definable in the structure $\str{\NN;+,\times}$. %(Hence the
word "arithmetical" is understood in the sense % usual in
mathematical logic, not in the sense from number theory.) An
arithmetical operation $f$ will be called strongest with
respect to the first order definability if addition and
multiplication  % $+$, $\times$ are first order definable in
the structure $\str{\NN,f}$. Examples of such operations are
presented, including polynomials of several variables (e.g.
$xy+1$, $x_1^2+x_2^2+x_3^2+x_4^2+x_5^2$, $x^7+y$) functions
related to divisibility and successor (e.g. $\gcd(x+1,y)$,
$\lcm(x+3,y+3)$) and pairing functions (i.e., bijections
$\NN^2\to\NN$). %They are chosen from the author's
(continuously actualized) %list \cite{Ko96}. Further,
uniform definability of binary relations from pairing
functions is proved.
