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Problems in Elementary Number Theory
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Problems in Elementary Number TheoryPeter Vandendriessche
Hojoo Lee
July 11, 2007
God does arithmetic. C. F. Gauss
,Chapter 1
Introduction
The heart of Mathematics is its problems. Paul Halmos
Number Theory is a beautiful branch of Mathematics. The purpose of this book is to present
a collection of interesting problems in elementary Number Theory. Many of the problems
are mathematical competition problems from all over the world like IMO, APMO, APMC,
Putnam and many others. The book has a supporting website at
http://www.problem-solving.be/pen/
which has some extras to offer, including problem discussion and (where available) solutions,
as well as some history on the book. If you like the book, you’ll probably like the website.
I would like to stress that this book is unfinished. Any and all feedback, especially about
errors in the book (even minor typos), is appreciated. I also appreciate it if you tell me about
any challenging, interesting, beautiful or historical problems in elementary number theory
(by email or via the website) that you think might belong in the book. On the website you
can also help me collecting solutions for the problems in the book (all available solutions will
be on the website only). You can send all comments to both authors at
peter.vandendriessche at gmail.com and ultrametric at gmail.com
or (preferred) through the website.
The author is very grateful to Hojoo Lee, the previous author and founder of the book, for
the great work put into PEN. The author also wishes to thank Orlando Doehring , who
provided old IMO short-listed problems, Daniel Harrer for contributing many corrections
and solutions to the problems and Arne Smeets, Ha Duy Hung , Tom Verhoeff , Tran
Nam Dung for their nice problem proposals and comments.
Lastly, note that I will use the following notations in the book:
Z the set of integers,
N the set of (strictly) positive integers,
N0 the set of nonnegative integers.
Enjoy your journey!
1
,Contents
1 Introduction 1
2 Divisibility Theory 3
3 Arithmetic in Zn 17
3.1 Primitive Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Quadratic Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Primes and Composite Numbers 22
5 Rational and Irrational Numbers 27
5.1 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6 Diophantine Equations 33
7 Functions in Number Theory 43
7.1 Floor Function and Fractional Part Function . . . . . . . . . . . . . . . . . . 43
7.2 Divisor Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.3 Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
8 Sequences of Integers 52
8.1 Linear Recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
8.2 Recursive Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8.3 More Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
9 Combinatorial Number Theory 62
10 Additive Number Theory 70
11 Various Problems 76
11.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
11.2 The Geometry of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
11.3 Miscellaneous problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
12 References 84
2
, Chapter 2
Divisibility Theory
Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful.
If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t
beautiful, nothing is. Paul Erdös
A 1. Show that if x, y, z are positive integers, then (xy + 1)(yz + 1)(zx + 1) is a perfect
square if and only if xy + 1, yz + 1, zx + 1 are all perfect squares.
Kiran S. Kedlaya
A 2. Find infinitely many triples (a, b, c) of positive integers such that a, b, c are in arithmetic
progression and such that ab + 1, bc + 1, and ca + 1 are perfect squares.
AMM, Problem 10622, M. N. Deshpande
A 3. Let a and b be positive integers such that ab + 1 divides a2 + b2 . Show that
a2 + b2
ab + 1
is the square of an integer.
IMO 1988/6
A 4. If a, b, c are positive integers such that
0 < a2 + b2 − abc ≤ c,
show that a2 + b2 − abc is a perfect square. 1
CRUX, Problem 1420, Shailesh Shirali
A 5. Let x and y be positive integers such that xy divides x2 + y 2 + 1. Show that
x2 + y 2 + 1
= 3.
xy
1 a2 +b2
This is a generalization of A3 ! Indeed, a2 + b2 − abc = c implies that ab+1
= c ∈ N.
3
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