Here are some elementary topics that come to mind

 - Continuous fractions  and  diophantine equations (linear and Pell equations).  (Check  Davenport's Higher arithmetic)
 - Geometry of numbers, Minkowski's theorem and applications
 -  Quadratic reciprocity 
 - Any one of Khincine's *Three pearls of number theory* (Dover Publications)
 - Transcendence of $e$ or $\pi$ (Felix Klein's   description of these proofs is  a gem.)
 -  Weyl's uniform distribution   theorem.