  Arithmetic differential equations are analogues of algebraic differential
equations in which derivative operators acting on functions are replaced by
Fermat quotient operators acting on numbers. Now, various remarkable
transcendental functions are solutions to algebraic differential equations; in
this note we show that, in a similar way, some remarkable transcendental
numbers (including certain "periods") are solutions to arithmetic differential
equations. Inspired by a recent paper of Manin, we then speculate on the
possibility of understanding the algebraic relations among periods via Galois
groups of arithmetic differential equations.
