A Selection of Problems in the Theory of Numbers focuses on mathematical problems within the boundaries of geometry and arithmetic, including an introduction to prime numbers. This book discusses the conjecture of Goldbach; hypothesis of Gilbreath; decomposition of a natural number into prime factors; simple theorem of Fermat; and Lagrange's theorem. The decomposition of a prime number into the sum of two squares; quadratic residues; Mersenne numbers; solution of equations in prime numbers; and magic squares formed from prime numbers are also elaborated in this text. This publication is a good reference for students majoring in mathematics, specifically on arithmetic and geometry.Table of ContentsOn the Borders of Geometry and ArithmeticWhat We Know and What We Do Not Know about Prime Numbers1. What are Prime Numbers?2. Prime Divisors of a Natural Number3. How Many Prime Numbers are There?4. How to Find All the Primes Less than a Given Number5. Twin Primes6. Conjecture of Goldbach7. Hypothesis of Gilbreath8. Decomposition of a Natural Number into Prime Factors9. Which Digits Can There Be at the Beginning and at the End of a Prime Number?10. Number of Primes Not Greater than a Given Number11. Some Properties of the n-th Prime Number12. Polynomials and Prime Numbers13. Arithmetic Progressions Consisting of Prime Numbers14. Simple Theorem of Fermat15. Proof That There is an Infinity of Primes in the Sequences 4k+1, 4k+3 and 6k+516. Some Hypotheses about Prime Numbers17. Lagrange's Theorem18. Wilson's Theorem19. Decomposition of a Prime Number into the Sum of Two Squares20. Decomposition of a Prime Number into the Difference of Two Squares and Other Decompositions21. Quadratic Residues22. Fermat Numbers23. Prime Numbers of the Form nn + 1, nnn + 1 etc.24. Three False Propositions of Fermat25. Mersenne Numbers26. Prime Numbers in Several Infinite Sequences27. Solution of Equations in Prime Numbers28. Magic Squares Formed from Prime Numbers29. Hypothesis of A. SchinzelOne Hundred Elementary but Difficult Problems in ArithmeticReferences