Selection of Problems in the Theory of Numbers (eBook)

Front Cover 1
A Selection of Problems in the Theory of Numbers 4
Copyright Page 5
Table of Contents 6
Acknowledgement 7
Chapter 1. ON THE BORDERS OF GEOMETRY AND ARITHMETIC 8
Chapter 2. WHAT WE KNOW AND WHAT WE DO NOT KNOW ABOUT PRIME NUMBERS 26
1. What are prime numbers? 26
2. Prime divisors of a natural number 27
3. How many prime numbers are there? 28
4. How to find all the primes less than a given number 29
5. Twin primes 31
6. Conjecture of Goldbach 33
7. Hypothesis of Gilbreath 35
8. Decomposition of a natural number into prime factors 36
9. Which digits can there be at the beginning and at the end of a prime number? 41
10. Number of primes not greater than a given number 43
11. Some properties of the nth prime number 44
12. Polynomials and prime numbers 46
13. Arithmetic progressions consisting of prime numbers 48
14. Simple Theorem of Fermat 49
15. Proof that there is an infinity of primes in the sequences 4k+1 4k+3 and 6k+5 53
16. Some hypotheses about prime numbers 57
17. Lagrange's theorem 59
18. Wilson's theorem 61
19. Decomposition of a prime number into the sum of two squares 64
20. Decomposition of a prime number into the difference of two squares and other decompositions 70
21. Quadratic residues 75
22. Fermat numbers 78
23. Prime numbers of the form nn+l, nnn+1, etc. 85
24. Three false propositions of Fermat 89
25. Mersenne numbers 90
26. Prime numbers in several infinite sequences 93
27. Solution of equations in prime numbers 95
28. Magic squares formed from prime numbers 96
29. Hypothesis of A. Schinzel 97
Chapter 3. ONE HUNDRED ELEMENTARY BUT DIFFICULT PROBLEMS IN ARITHMETIC 99
REFERENCES 124