  Every polynomial $P(X)\in \mathbb Z[X]$ satisfies the congruences
$P(n+m)\equiv P(n) \mod m$ for all integers $n, m\ge 0$. An integer valued
sequence $(a_n)_{n\ge 0}$ is called a pseudo-polynomial when it satisfies these
congruences. Hall characterized pseudo-polynomials and proved that they are not
necessarily polynomials. A long standing conjecture of Ruzsa says that a
pseudo-polynomial $a_n$ is a polynomial as soon as $\limsup_n \vert
a_n\vert^{1/n}<e$. Under this growth assumption, Perelli and Zannier proved
that the generating series $\sum_{n=0}^\infty a_n z^n$ is a $G$-function. A
primary pseudo-polynomial is an integer valued sequence $(a_n)_{n\ge 0}$ such
that $a_{n+p}\equiv a_n \mod p$ for all integers $n\ge 0$ and all prime numbers
$p$. The same conjecture has been formulated for them, which implies Ruzsa's,
and this paper revolves around this conjecture. We obtain a Hall type
characterization of primary pseudo-polynomials and draw various consequences
from it. We prove that any primary pseudo-polynomial with an algebraic
generating series is a polynomial. We make the Perelli-Zannier Theorem
effective and show that its conclusion does not necessarily hold if $\limsup_n
\vert a_n\vert^{1/n}\le e$. We prove a P\'olya type result: if there exists a
function $F$ analytic in a right-half plane with not too large exponential
growth (in a precise sense) and such that for all large $n$ the primary
pseudo-polynomial $a_n=F(n)$, then $a_n$ is a polynomial. Finally, we show how
to construct a non-polynomial primary pseudo-polynomial starting from any
primary pseudo-polynomial generated by a $G$-function different of $1/(1-x)$.
