Lecture 4 - Prime Numbers; The Fundamental Theorem of Arithmetic

Summary

Today we spent the first half of the class exploring questions about prime numbers. Along the way we proved that there are infinitely many prime numbers and that there are arbitrarily large gaps between prime numbers. We also saw a formula which gives a rough count for the number of integers up to a given number x, and we saw some conjectures about other behaviors about prime numbers. In the last half of the class we started a proof of the Fundamental Theorem of Arithmetic.

Asking Questions About Primes

Having actually gone through and talked about the basics regarding prime numbers, we now begin to wonder what can be said about primes. Here are a few basic questions you might want to know

  • how many primes are there? for instance, is the number of primes finite?
  • if the number of primes isn't finite, do we at least have a reasonable guess as to how many primes there are of a given magnitude?
  • do we know how "spread out" the prime numbers are? do they come in clusters, or should we expect that they are always far apart from each other?
  • is there a formula which allows us to quickly generate prime numbers?
  • do prime numbers obey any special properties? for instance, are there more primes which leave remainder 1 after division by 4 than there are primes which leave remainder 3 after division by 1?
  • what can you say about how the primes behave under addition?

These are all good questions, and some of them have nice, easy answers. Alternatively, some of these questions are exceedingly difficult to investigate. We'll cover a sampling now.

The Infinitude of Primes

The question on the number of primes has been around for a long time, and the answer was known at least two thousand years ago. Here's the proof that Euclid gave in his Elements.

Theorem: There are infinitely many primes.

Proof: Again, we'll proceed by contradiction: assuming there are finitely many primes, massaging this condition into a contradiction, and then concluding that a finite number of primes is impossible.

So suppose you have a list of all the prime numbers, and call them $p_1, p_2, \cdots, p_r$. Then we'll form the integer $N = p_1 p_2 \cdots p_r + 1$. Notice that for any $p_i$ in our list of primes, we cannot have $p_i \mid N$; if we did, then we'd also know that

(1)
\begin{align} p_i \mid N - p_1p_2 \cdots p_r = 1. \end{align}

But we know that N has to have at least 1 prime factor p. Since this prime number isn't one of the primes in our list, we conclude that the list of primes we started off with was incomplete. $\square$

Gaps and Clusters in Primes

Now that we know there are infinitely many primes, we might want to have a reasonable idea of how the primes are spaced out amongst the integers. Displaying their typical quirkiness, the answer to this question seems to be on both extremes: some primes have wide gaps to their next neighbor, while — conjecturally, at least — others are as close as can be.

On the one extreme, we have a theorem which tells us that large gaps between primes numbers are known to exist.

For any positive integer M, there is a string of at least M consecutive composite integers.

Proof: The M integers between

(2)
\begin{align} (M+1)!+2, (M+1)! + 3, \cdots, (M+1)!+M+1 \end{align}

are all composite, since the first is divisible by 2, the second by 3, etc. $\square$

On the other hand, empirical evidence suggests that there are also lots of primes which are quite close to each other. The most famous result in this vein is

The Twin Prime Conjecture: There are infinitely many primes p such that p+2 is also prime.

For those who are interested, the record largest twin primes to date can be found at The Largest Known Primes Page; as of this morning, the largest twin primes were

(3)
\begin{align} 2003663613\cdot 2^{195000}\pm 1, \end{align}

two numbers which have something like 60,000 digits.

The Prime Number Theorem

With all the spreading out and bunching up between the prime numbers, one might think that it would be hard to give an estimate for the number of primes of a given magnitude. However, one of the biggest results in number theory — and one which is almost always proved using techniques from complex analysis (!) — tells us exactly this information. It uses a function $\pi(x)$, which is defined as the number of primes less than or equal to a given number x. (So, for instance, we have $\pi(11) = 5$ since the primes less than or equal to 11 are $\{2,3,5,7,11\}$.

The Prime Number Theorem: $\frac{\pi(x)\ln(x)}{x} \to 1$ as $x \to \infty$.

This says that for large values of x, the number of primes less than or equal to x is about $\frac{x}{\ln(x)}$.

Primes of a particular form

Now that we know a little bit about primes, it is natural to ask: how can we go about finding them? The answer to this question, sadly, is that there's not really a general method for finding all primes aside from ''brute force'' techniques like our sieve method. Indeed, the difficulty in finding primes is one of the hard problems which helps keep our world afloat right now: encryption online is dependent on the fact that it's really hard to factor large numbers.

Even though it's hard to come up with an exhaustive list of all primes, there are some places where prime hunters go to search for big game. Although finding large primes was a kind of pleasant amusement amongst mathematicians a hundred years or so ago, today it is big business: the aforementioned internet security applications of primality require large primes to work. Hopefully we'll be able to talk about all this more at the end of the term.

Mersenne Primes

The largest primes found these days all happen to take a particular form: they can be expressed as $2^p-1$ for a prime number p. These are the so-called Mersenne Primes. There was a recent development (i.e., early last semester), when the Great Internet Mersenne Prime Search (GIMPS) came across the new largest prime number. This number is

$2^{43112609}-1$

and has around 13 million digits. If you want, you can use your computer to help GIMPS out; maybe it will be your computer which finds the next largest prime!

The Primes Under Addition

Finally, we consider the question: what happens when you add together prime numbers? Since primes are defined based upon a multiplicative property, one might not expect that they really have a lot of interesting additive structure. It seems, however, that they have a very rich additive structure. For instance, here's a long-standing conjecture about how the primes behave under addition:

Goldbach's Conjecture: Every even integer at least 4 can be expressed as the sum of two prime numbers.

Though plenty of smart people have been thinking about this problem for a couple of hundred years, and although it has been verified for "lots" of even numbers (can someone post to the Wiki how many even integers have been verified to satisfy this condition?), no one has yet been able to prove that it is true.

The Fundamental Theorem of Arithmetic

Having covered many of the basics, it's now time for us to knock down the Fundamental Theorem of Arithmetic. This theorem is something which you all have seen many times before — whether explicitly or not — and is an incredibly useful tool in number theory.

We need a preliminary lemma before we can knock down the Fundamental Theorem. This preliminary result is known as Euclid's Lemma, and it is essentially a special case of one of your homework problems for the week (44a).

Euclid's Lemma: If p is a prime number and $p \mid ab$, then either $p \mid a$ or $p \mid b$.

Proof: Suppose that $p \nmid a$, and we'll argue that $p \mid b$. For this, notice that $p \nmid a$ forces $(p,a) = 1$ — the only divisors of p are 1 and itself, and we already know that p isn't a divisor of a. Applying 44a from your homework gives the desired result. $\square$

The Proof of the Fundamental Theorem

We're now ready to prove the Fundamental Theorem of Arithmetic. Recall that it says

The Fundamental Theorem of Arithmetic - Every positive integer at least 2 can be uniquely expressed as a product of prime numbers.

We'll break our proof into two parts

  1. Existence: that every $n > 1$ can be written as $n = p_1\cdots p_r$ for some prime numbers $p_i$
  2. Uniqueness: there is only one such way to factor a given integer

We only had time in class to cover the first statement; we'll prove the second in class on Monday.

Existence: Suppose that there were integers greater than 1 which couldn't be factored into a product of primes. This would mean that there is a smallest such integer (by the well-ordering principle), and we'll call this smallest element n. Now n can't be prime since otherwise n is already an expression of itself as a product of primes. Hence $n = ab$ for some $1 < a,b < n$. Since both a and b are smaller than n, this means that they must be elements which do have prime factorizations (since n was selected as the smallest positive integer which didn't have this property). Therefore $a = p_1 \cdots p_k$ and $b = q_1 \cdots q_r$ for appropriate primes $p_i,q_j$. But then we have

(4)
\begin{align} n = ab = p_1\cdots p_k q_1 \cdots q_r, \end{align}

a prime factorization of n. Since this contradicts the selection of n as the least element without a prime factorization, we must conclude that every integer greater than 1 can be factored as a product of primes.

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