## A Collection of Problems in The Theory of Numbers – Sierpinski

In this post, we will see the book A Collection of Problems in The Theory of Numbers by Waclaw Sierpinski. 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.

The book was translated from Polish by A. Sharma and was published in  1964.

You can get the book here.

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# Contents

On the Borders of Geometry and Arithmetic

What We Know and What We Do Not Know about Prime Numbers

1. What are Prime Numbers?

2. Prime Divisors of a Natural Number

3. How Many Prime Numbers are There?

4. How to Find All the Primes Less than a Given Number

5. Twin Primes

6. Conjecture of Goldbach

7. Hypothesis of Gilbreath

8. Decomposition of a Natural Number into Prime Factors

9. 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 Number

11. Some Properties of the n-th Prime Number

12. Polynomials and Prime Numbers

13. Arithmetic Progressions Consisting of Prime Numbers

14. Simple Theorem of Fermat

15. Proof That There is an Infinity of Primes in the Sequences 4k+1, 4k+3 and 6k+5

16. Some Hypotheses about Prime Numbers

17. Lagrange’s Theorem

18. Wilson’s Theorem

19. Decomposition of a Prime Number into the Sum of Two Squares

20. Decomposition of a Prime Number into the Difference of Two Squares and Other Decompositions

22. Fermat Numbers

23. Prime Numbers of the Form nn + 1, nnn + 1 etc.

24. Three False Propositions of Fermat

25. Mersenne Numbers

26. Prime Numbers in Several Infinite Sequences

27. Solution of Equations in Prime Numbers

28. Magic Squares Formed from Prime Numbers

29. Hypothesis of A. Schinzel

One Hundred Elementary but Difficult Problems in Arithmetic

References 