Preface

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Number theory is concerned with properties of the integers: \[\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots.\nonumber\] The great mathematician Carl Friedrich Gauss called this subject arithmetic and of it he said:

Mathematics is the queen of sciences and arithmetic the queen of mathematics."

At first blush one might think that of all areas of mathematics certainly arithmetic should be the simplest, but it is a surprisingly deep subject.

We assume that students have some familiarity with basic set theory, and calculus. But very little of this nature will be needed. To a great extent the book is self-contained. It requires only a certain amount of mathematical maturity. And, hopefully, the student’s level of mathematical maturity will increase as the course progresses.

Before the course is over students will be introduced to the symbolic programming language Maple which is an excellent tool for exploring number theoretic questions.

If you wish to see other books on number theory, take a look in the QA 241 area of the stacks in our library. One may also obtain much interesting and current information about number theory from the internet. See particularly the websites listed in the Bibliography. The websites by Chris Caldwell and by Eric Weisstein are especially recommended. To see what is going on at the frontier of the subject, you may take a look at some recent issues of the Journal of Number Theory which you will find in our library.

Here are some examples of outstanding unsolved problems in number theory. Some of these will be discussed in this course. A solution to any one of these problems would make you quite famous (at least among mathematicians). Many of these problems concern prime numbers. A prime number is an integer greater than 1 whose only positive factors are 1 and the integer itself.

(Goldbach’s Conjecture) Every even integer \(n > 2\) is the sum of two primes.
(Twin Prime Conjecture) There are infinitely many twin primes. [If \(p\) and \(p+2\) are primes we say that \(p\) and \(p+2\) are twin primes.]
Are there infinitely many primes of the form \(n^2+1\)?
Are there infinitely many primes of the form \(2^n - 1\)? Primes of this form are called Mersenne primes.
Are there infinitely many primes of the form \(2^{2^n} +1\)? Primes of this form are called Fermat primes.
(\(3n+1\) Conjecture) Consider the function \(f\) defined for positive integers \(n\) as follows: \(f(n) = 3n+1\) if \(n\) is odd and \(f(n)=n/2\) if \(n\) is even. The conjecture is that the sequence \(f(n), f(f(n)), f(f(f(n))), \cdots\) always contains 1 no matter what the starting value of \(n\) is.
Are there infinitely many primes whose digits in base 10 are all ones? Numbers whose digits are all ones are called repunits.
Are there infinitely many perfect numbers? [An integer is perfect if it is the sum of its proper divisors.]
Is there a fast algorithm for factoring large integers? [A truly fast algoritm for factoring would have important implications for cryptography and data security.]

Famous Quotations Related to Number Theory

Two quotations from G. H. Hardy:

In the first quotation Hardy is speaking of the famous Indian mathematician Ramanujan. This is the source of the often made statement that Ramanujan knew each integer personally.

I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways. ”

Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.

Two quotations by Leopold Kronecker

God has made the integers, all the rest is the work of man.

The original quotation in German was Die ganze Zahl schuf der liebe Gott, alles Übrige ist Menschenwerk. More literally, the translation is “ The whole number, created the dear God, everything else is man’s work.” Note in particular that Zahl is German for number. This is the reason that today we use \(\mathbb{Z}\) for the set of integers.

Number theorists are like lotus-eaters – having once tasted of this food they can never give it up.

A quotation by contemporary number theorist William Stein:

A computer is to a number theorist, like a telescope is to an astronomer. It would be a shame to teach an astronomy class without touching a telescope; likewise, it would be a shame to teach this class without telling you how to look at the integers through the lens of a computer.