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.