## Mahler-Lech Theorem

Let be a Field of Characteristic 0 (e.g., the rationals ) and let be a Sequence of elements of which satisfies a difference equation of the form

where the Coefficients are fixed elements of . Then, for any , we have either for only finitely many values of , or for the values of in some Arithmetic Progression.

The proof involves embedding certain fields inside the p-adic Number for some Prime , and using properties of zeros of Power series over (Strassman's Theorem).