Abstract: One considers an ordinary linear differential equation $L(y)=0$ with $L(y):=y^{(n)}+a_{n-1}y^{(n-1)}+\cdots +a_1y^{(1)}+a_0y$ having coefficients in, say the field ${\bf Q}(z)$. Grothendieck conjectured that all solutions of $L(y)=0$ are algebraic if and only if for almost all primes $p$ the reduction of $L$ modulo $p$ has ``enough'' solutions in the field ${\bf F}_p(z)$. We will investigate this conjecture for $n=1,2$ and explain the various connections with number theory.