Proofs from THE BOOK


How much do I want to read more? 8/10

What a superb topic, to filter the whole spectrum of mathematics with extracting the golden proofs.
While it says we can understand everything with undergraduate mathematics (high school?), reading the proof doesn't look so obvious.


Preface

all we offer here is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations.

this book is supposed to be accessible to readers whose backgrounds include only a modest amount of technique from undergraduate mathematics. A little linear algebra, some basic analysis and number theory, and a healthy dollop of elementary concepts and reasonings from discrete mathematics should be sufficient to understand and enjoy everything in this book.

Number Theory

Chapter 1 - Six proofs of the infinity of primes

It is only natural that we start these notes with probably the oldest Book Proof, usually attributed to Euclid (Elements IX, 20). It shows that the sequence of primes does not end.

Euclid’s Proof.

For any finite set {p1,…,pr} of primes, consider the number n = p1p2 ···pr + 1.
This n has a prime divisor p. But p is not one of the pi: otherwise p would be a divisor of n and of the product p1p2 ···pr, and thus also of the difference n − p1p2 ···pr = 1, which is impossible.
So a finite set {p1 , . . . , pr } cannot be the collection of all prime numbers.

N = {1, 2, 3, . . .} is the set of natural numbers
Z = {…,−2,−1,0,1,2,…} the set of integers
P = {2,3,5,7,…} the set of primes.

The natural numbers grow beyond all bounds, and every natural number n ≥ 2 has a prime divisor.
These two facts taken together force P to be infinite.

The next proof is due to Christian Goldbach (from a letter to Leon- hard Euler 1730),
the third proof is apparently folklore,
the fourth one is by Euler himself,
the fifth proof was proposed by Harry Fürstenberg,
the last proof is due to Paul Erdos.