3 Number Theory

Number theory is celebrated by mathematicians as a pure form of abstract thought, a

distillation of reason. Carl Friedrich Gauss called it the “queen of mathematics,” while

G. H. Hardy, in A Mathematician’s Apology, admired its pure, abstract isolation, praising

the fact that it was unencumbered by the physical world, without use or application.

And yet, despite this, in a strange and surprising twist of fate, the theory has in contem-

porary times found key practical applications; deep number-theoretic ideas, for example,

lie at the core of cryptography and internet security. Our dreamy iridescent theory of

numbers, it turns out, supposedly without use or application, does in fact have important

applications, vital for commerce and communication, so much so that number-theoretic

ideas are now firmly established via cryptography in the foundations of our economy.

3.1 Prime numbers

Let us develop some elementary number theory, beginning with the prime numbers. What

is a prime number? At this question, I imagine, perhaps a nervous laugh goes through the

classroom—of course we are all familiar with prime numbers, right? A helpful student

suggests, tentatively, that a number is prime if the only divisors of the number are 1 and

itself. Is this a good definition? Let us quibble. One minor issue is that numbers can

have negative divisors. We want 3 to be prime, to be sure, but does the factorization

3 = (−3) · (−1) mean that it has −3 and −1 also as divisors? Let us therefore interpret

the student’s proposal to refer only to positive divisors. Similarly, we do not want

7

2

· 2to

rule out 7 as prime, and so we should interpret the student’s proposal as referring only to

positive integer divisors.

A more serious issue, however, is the question of whether the number 1 should count as

prime. Is 1 prime? Since 1 has no positive integer divisors other than 1 and itself, it would

seem that, according to the student’s proposal, we would say that, indeed, 1 is prime.

Nevertheless, this would go against the advice of many mathematicians, who have come to

the conclusion that we do not actually want to include 1 amongst the prime numbers. On

this view, the student’s suggestion is not quite exactly right.