transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
For $p$ a prime number, then the Fermat quotient of any integer $a$ is the quotient
of the difference between the $p$th power of $a$ and $a$ itself by $p$. By Fermat's little theorem this is indeed an integer, i.e. conversely one has for all $a\in \mathbb{Z}$ that
for a uniquely defined integer $\partial_p a$.
(The $p$-power operation $(-)^p$ here is the one that restricts to the Frobenius homomorphism after p-localization and the one which is a shadow of the power operations in E-infinity arithmetic geometry (Lurie, remark 2.2.7).)
As the notation is meant to suggest, the Fermat quotient as a map $\mathbb{Z} \to \mathbb{Z}$ from the (underlying set of the) ring of integers to itself is analogous to a derivation. It is not quite an ordinary derivation, but satisfies conditions of what has been called a $p$-derivation.
In view of this, in the context of arithmetic differential equations the Fermat quotient is interpreted as an analog in arithmetic geometry of actual derivations in algebraic geometry/complex analytic geometry. For more on this see also at Borger's absolute geometry the section Motivation.