Chapter I. The adventure begins.
Contents: 1. Daddy, what's the biggest number?
2. What are the prime numbers, and what do they have to do with space aliens?
3. There ain't no biggest prime neither.
4. A brain like a sieve.
5. Twin primes, and proofs Sudoku style.
6. Your cellphone is crawling with prime numbers.
1. Daddy, what's the biggest number?
Anyone who's been around small children much has probably heard a
question like this one. Your first answer is probably ``actually there
isn't a biggest number'', to which the child will likely reply ``why
not?''. Well, you say, suppose you had any enormous number, like a
gazillion. If you just add one to it you get a gazillion and one, which
is even bigger. So no matter how big a number you take---the number of
stars in the sky, the number of grains of sand on the beach---you can
always add one to make it bigger.
Another way to say it (not recommended when explaining things to a little
person) is that there are infinitely many whole numbers 1,2,3,..., or
``positive integers'' as we like to call them. So it's a subtle concept,
involving as it does the notion of infinity. In any case, your
explanation to the child is a simple example of a ``proof by
contradiction''. A bit more formally, it would go like this:
I claim there is no biggest number. Proof: Suppose (in order to derive a
contradiction) that there was a biggest number. Let's call this
presumptuous number ``N''. (You could call it ``Finley'' or ``Hannah'' or
``thingamagig'', but N is short and sweet and serves to remind us that it
stands for a number.) So, Mister N, you claim to the biggest number?
Hah! I'll just add one to you to get a bigger number N+1. This
contradiction proves you are a fraud, a liar, and a menace to
society. No ``biggest number'' exists!
This might seem like making a mountain out of a molehill. But it's
always good to warm up, to stretch out your brain a bit before moving
on, so as to avoid blowing out any neurons. We'll use a more subtle
version of the same idea later to prove there is no biggest prime
number.
2. What are the prime numbers, and what do they have to do with space
aliens?
Let's think about some things you can do with numbers. You can add them,
for instance, or you can multiply them. Let's indicate multiplication
with an asterisk: 2*3=6, 3*5=15, etc. (I do this only because of the
limited symbols available on this blog page). One interesting
observation is that some numbers can be ``factored'' and others
cannot. For example 15=3*5, but there is no way to factor 7 other than
the trivial factorization 7=1*7. This state of affairs even has a nice
geometric interpretation: If you want to put 15 chocolates in a box you
can arrange them in three rows of five each:
x x x x x
x x x x x
x x x x x
But you can't do it with 7, other than by putting them all in one boring
row; no matter how you do it there are rows of different lengths, for
instance.
x x x x
x x x
So we've discovered some very special and interesting numbers, namely
those numbers that cannot be factored into smaller numbers. If we were
the first to discover these numbers we could name them whatever we like.
We could call them ``thin numbers'', ``groovy numbers'', or ``apple
fritters'' if we so desired. But we're a few millenia late to have
naming rights, so we'll stick to the standard term ``prime number'' or
simply ``prime''. While we're at it, though, it behooves us to define
our terms precisely. The description we just gave ``numbers that can't
be factored into smaller numbers'' is precise enough for our purposes,
but if taken literally it includes the number 1 as a prime. There is
nothing wrong from a logical standpoint with counting 1 as a prime, but
if you do so you'll soon discover that to say anything interesting about
primes you'll constantly have to insert disclaimers such as ``suppose p
is a prime other than 1''. Experience shows that it's better to just exclude
it by fiat. So here's our official definition:
A prime number is a number greater than 1 that cannot be factored into
smaller numbers.
You can (and should!) check that the primes less than 30 are the
following:
2,3,5,7,11,13,17,19,23,29
Except for 2, all prime numbers are odd. Why?
In one of my favorite movies, ``Contact'' (based on the book by Carl
Sagan and starring Jodie Foster), beings from beyond our galaxy send out
radio pulses in prime groups: first two, then three, then five and so
on. The point is that prime-ness is an intrinsic property of numbers,
completely independent of what symbols are used to denote them, whether
you use base ten as we do, what language you speak, and so on. The
extra-galactic reasoning therefore is that any intelligent form of life
will recognize the sequence of primes and realize that another
intelligent life-form is attempting to make contact.
Prime numbers are important because they are the building blocks for
making all whole numbers, via multiplication. If you take any such
number greater than one, you can factor it as a product of prime
numbers. For example, 30=2*3*5, 72=2*2*2*3*3.
But hold on just a doggone minute! Giving a couple of examples doesn't
prove anything. How do we know that ALL numbers admit such a
factorization into primes?
Well, suppose someone walks up to you on the street and hands you a
positive integer. Let's call this number N. If N happens to be a prime
number, then N is its own factorization (with just one factor). If it
isn't prime, then N can be factored into two smaller numbers N=a*b. If a
and b are primes, then we have our factorization. If not, then they in
turn can be factored into smaller numbers. If we continue this process,
it has to stop after a finite number of steps, because positive integers
(just good old whole numbers, remember) can't keep getting smaller
forever. When the process stops, we have written N as a product of
smaller numbers, none of which can be factored further and so are primes
by definition. Here ends the (convincing, I hope) explanation.
Let's illustrate with a specific example. Say you started with
N=60. There are multiple ways you might do the first step, but probably
the most obvious would be 60=6*10. Neither 6 nor 10 is prime, so we
factor them further: 6=2*3, 10=2*5. We luck out and are done after two
steps: 60=2*3*2*5 is our prime factorization. However, and this is a
crucial point, an example does not constitute a proof. A proof is a
convincing argument that works in ALL cases under consideration. It
should hold up in a court of law under rigorous cross-examination. The
person-on-the-street argument given above is an example of a proof. In
a textbook you might see something like this:
Theorem. Every number greater than 1 can be factored into primes.
Proof: (Insert here a cleaned-up, more precisely worded version of the
version above.)
There's nothing wrong with this style, but some people are put off
and/or intimidated by the words ``Theorem'' and ``Proof''. So if you
prefer, rewrite it as follows:
Cool Fact: Every number greater than 1 can be factored into primes.
Here's why it's true, dude: Suppose someone walks up to you on the
street...etc.
Also important to note is that we didn't START by writing down Theorem:
blah blah blah Proof: blah blah blah. We started with a question. We
found an answer and convincingly demonstrated its truth. The more formal
``theorem'' and ``proof'' come later, after the smoke has cleared and
the dust has settled. The reason that we mathematicians often write out
our proofs very pedantically, dotting every i and crossing every t, is
that the deeper the mathematics, the easier it is to make subtle
mistakes. Very careful reasoning is required. But that's not the form in
which the proof is discovered.
Note also the following corollary of the theorem.
Corollary: If N is bigger than 1, then there is at least one prime that
divides N.
(By ``divides N'' I mean the same thing as ``is a factor of N''. In
mathematics, as in other languages, there are often different words for
the same thing.)
In mathematics a ``corollary'' of a theorem is a second theorem (cool
fact!) that follows easily from the first. What constitutes ``easily''
is of course a matter of opinion, but in the present case it's like,
totally easy: The theorem says N can be factored into primes, and any of
these prime factors will divide it.
3. There ain't no biggest prime neither.
What's the biggest prime? This would be good to know, as then we'd have
a finite list of the primes, the basic building blocks for all numbers.
Well, there ain't no such thing. The Greek mathematician Euclid (who
lived around 300 BC and is most famous for his work in geometry), found
a clever argument to prove that there are infinitely many primes. His
proof is by contradiction. Such a proof really is like a
cross-examination. Imagine that Mr. P is on trial for mathematical
fraud, having sold stock based on his claim to be the largest prime
number. The defense has called him as a witness, and Euclid, the
prosecuting attorney, now proceeds to the cross-examination.
Euclid: Mr. P, you claim to be the largest prime, is that correct?
P (with a disgusted look): Yes.
Euclid: I see. (He pauses, shuffling a few papers.) Now Mr. P, suppose I
multiply together the first few primes---let's say 2,3 and 5, and then
add 1. Is this number divisible by 2, 3, or 5?
Defense: Objection, your honor! Irrelevant!
Judge: Overruled. I'll give you some leeway, Mr. Euclid, but this better
be leading somewhere. The witness may answer the question.
P (sarcastically, as though talking to a child): Well, 2 times 3 times 5
is 30, and if I add one I get 31. That isn't divisible by 2, 3 or 5. If
that was too fast for you I can go over it again.
Euclid: All right, but isn't it true that you really didn't need to do
the multiplication at all? I mean, if you multiply 2,3 and 5 together,
then without even writing down the answer you know that adding 1 will
produce a number whose remainder, after division by 2,3 or 5, is 1.
Defense: Objection! I fail to see what this has to do with---
Judge: Sit down, counselor! I'll allow it, but I warn you, Mr. Euclid,
that my patience is wearing thin.
P: Well yeah, sure. I could take 2,3,5,7,11, multiply them together,
add 1 and know in advance that if I divide any of those five primes into the
new number the remainder will be 1. That's how remainders work. But
maybe they didn't cover that when you went to school.
Euclid (continuing to ignore P's sarcasm): So if you're the biggest
prime, there is a finite list of ALL primes and we can multiply them all
together. Call this number N. Then N+1 is not divisible by any of the
aforementioned primes, because if you divide by any prime on the list
you'll get a remainder of 1.
P: Right. So what?
Euclid: I call the attention of the court to the earlier testimony of
expert witness Herr Professor Doctor Mitchell, who testified that for
any number bigger than 1, there is SOME prime that divides it. So there
is a prime---I'll call it q, if it please the court---such that q
divides N+1. In other words, the remainder after division by q is 0, not
1. So this prime q is not on the list, it is bigger than you, and your
claim is a lie!
(Pandemonium erupts in the court. Mr. P is found guilty and sentenced to
a year of watching Dragnet re-runs.)
More formally, say for publication in the Proceedings of the North
Kirkland Philosophical Society, one might write things out like so:
Theorem. There are infinitely many primes.
Proof: Suppose (in order to reach a contradiction) that the theorem is
false, i.e. that there are only finitely many primes. Multiply them all
together (this is possibile because there are only finitely many); call
the resulting number N. Then dividing any of the primes into N+1
produces a remainder of 1, so no prime divides N+1. But since N+1 is
greater than 1, there IS a prime that divides. We have therefore arrived
at a contradiction, proving that the theorem is true.
4. A brain like a sieve.
When I was a boy, my father used to say I had a ``brain like a sieve'',
a reference to the absent-mindedness that apparently was already
evident. Even sieves can be put to good use, however.
To expand our above list of primes more efficiently, we can use a method
picturesquely known as the ``Sieve of Eratosthenes''. Eratosthenes (276
BC-195BC) was a Greek man of many talents---mathematician, poet,
``Father of Geography''. The Wikipedia article on him includes an
color-coded animation of the sieve being used to find all primes less
than 121. But let's see how you might discover this method yourself to
find all primes below 121. Start by writing down all the numbers from 2
to 120. Or at least imagine that you've done so; as you'll see, there
are many shortcuts to speed up the process.
1. Cross out all even numbers greater than 2 on your list, since these
have 2 as a factor and so can't be prime. This is superty-duperty easy,
as Kaia would say, because in our customary base 10 a number is even if
and only if its last digit is even. But hey, even without knowing this,
you could just start at 4 and cross out every other number 4,6,8,....
2. Cross out all numbers with 3 as a factor, other than 3 itself. So you
cross out 6, 9, 12, 15,...ah, but actually all the even ones 6, 12, 18
are already gone.
3. Skip to the next number that hasn't yet been crossed out. It has to
be a prime (why?), in this instance 5. Cross out all multiples of 5,
which is very easy in base 10 because they are the numbers whose last
digit is 0 or 5 (why?). All such multiples that also multiples of 2 or 3
have already been crossed out, speeding up the process further.
4. Repeat step 3. The next number not already crossed out is 7, which
indeed is prime as claimed. Proceed as before to cross out all multiples
of 7 (there won't be many left).
5. Repeat step 3. The next number not already crossed out is 11...but
wait a minute, we're already done! There's no need for Step 5! We know
without even looking that every multiple of 11 has already been crossed
out. Do you see why? (Hint: 121=11*11.) If you can answer this then you
see why we can stop the process at Step 4. All remaining numbers on the
list are primes!!
For future reference, here's the list of all primes below 121:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,
107, 109, 113.
5. Twin primes, proofs Sudoku style, and other diversions.
If you ponder the above list of primes, you notice some things about the
way they're ``distributed''. For example, every now and then one
encounters ``twin primes'', meaning two consecutive odd numbers that are
both prime, such as 3,5 and 29,31. Taking our cue from Euclid, we ask:
Are there infinitely many such pairs of twin primes? No one knows! This
is an example of an unsolved problem in mathematics. If you can answer
it you will be famous, and probably will rake in some big bucks too.
Okay, what about ``triplet primes'', meaning three consecutive odd
numbers that are prime, such as 3,5,7? Staring at the list above, you'll
see that 3,5,7 are the only triplet primes below 121. It's a fun problem
to prove that these are the ONLY triplet primes. I'll give some hints
after a digression on Sudoku (which you can skip if you're not familiar
with the game).
When you play Sudoku, you are doing mathematical proofs. The proofs in
question could be styled ``proofs by contradiction'' or ``proofs by
process of elimination''. In the simplest case it works something like
this: Maybe you have a row with only three squares remaining to be
filled in, so that you know that one of the squares has to be, let's
say, a 3,5 or 8. But that square already has a 3 in its column, and an 8
in its 3 by 3 block. So by process of elimination you have PROVED that
it has to be a 5. Sudoku is pure mathematics!
Now, back to the triplet prime problem. Usually I don't like to give
hints, because they interfere with your creativity. This is true at all
levels of mathematics. But since we're just getting started, and you
might still be uncertain about what a ``proof'' is, I'll make some
suggestions. First of all, experiment with small numbers to get an idea
of what's going on. Examples don't constitute a proof, but they can
SUGGEST a proof. For instance, 5,7,9 aren't triplet primes because 9
isn't prime; it's divisible by 3. Nor does 13, 15, 17 work, as 15 isn't
prime (it's divisible by 3). Or try 27, 29, 31. Nope; 27 is divisible by
3. In fact it appears that in every case, one number in the triplet
is divisible by 3. This suggests proceeding as follows: We need names
for our triplets; let's call the smallest one n. Then the other two are
n+2 and n+4. Try proving that one of the numbers n,n+2,n+4 has to be
divisible by 3 and therefore can't be prime (except in the special case
n=3). Here's where ``Sudoku style'', or process of elimination, comes
in. If n or n+2 is divisible by 3 we can quit; our theorem is proved. So
suppose that n and n+2 are NOT divisible by 3 and prove that this forces
n+4 to be divisible by 3. Further hints available on request!
Here's another interesting question, at the opposite extreme from twin
primes. How big can the gaps between consecutive primes be? If you look
at the list of primes below 121 given above, you'll see that for a long
time there is no gap bigger than 6, e.g. 23 and 29, and then we suddenly
get a gap of 8 from 89 to 97. In fact the gaps can be arbitrarily large.
Once you have the right creative idea, the proof of this fact is
surprisingly ``easy''. But of the course that little idea is the catch!
Here it is: Let's use the traditional, centuries-old notation n! to mean
the product of all the numbers from 1 to n. Thus
3! =1*2*3=6
4! =1*2*3*4=24
5!= 1*2*3*4*5=120
and so on. I don't know who first came up with the exclamation point
notation, but it was long ago. You could just as well use a heart, a
smiley-face, or a skull-and-crossbones, but n! is the standard (to be
read as ``n factorial''). Now consider the following list of
consecutive numbers:
n!+2, n!+3, n! +4,...,n!+n.
Here the ellipses ... are a standard way of saying ``keep going in this
way until you get to n!+n''. Then no number in this list is prime. Do
you see why? Prove this to yourself! There are n-1 numbers on the list,
so the gap between the biggest prime less than n!+2 and the next prime
is at least n-1. Since n can be any number here---a thousand, a million,
a gazillion, whatever---this proves that there can be arbitrarily large
gaps between consecutive primes.
The distribution of primes has been intensively studied, and is
currently a very active area of research. The biggest unsolved problem
in this area, often regarded as the biggest unsolved problem in
mathematics, is the ``Riemann hypothesis''. It was first conjectured by
the brilliant German mathematician Bernhard Riemann (1826-1866). The Clay
Research Institute has offered a million dollars for its solution.
(Unfortunately, I can't explain it here; you'd need a solid background
in undergraduate mathematics.)
6. Your cellphone is crawling with prime numbers.
In his 1940 essay ``A mathematician's apology'' (the word ``apology'' is
used in the sense of justification), the prominent British mathematician
G.H. Hardy (1877-1947) claimed that number theory would never have any
practical applications. Number theory was his field, and he regarded the
lack of applications as a good thing, as then it could never be applied
to war. Within a generation or two he was proved spectacularly wrong, as
number theory---this includes the prime numbers we've been
discussing---is absolutely crucial for our current technology: the
internet, CD's, DVD's, cellphones, anything to do with computers and
data transmission. There is an ``elliptic curve'' in your cellphone, for
example, and it is crawling with prime numbers.
The main applications that I'm aware of (and I've only scratched the
surface) have to with (1) encryption, and (2) error-correcting
codes. Secure data transmission---of your bank information or your
credit card number, of military secrets---requires that it be
encrypted. An ingenious way of doing this using prime numbers was
invented about thirty years after Hardy's statement that number theory
would never be applied. Later a colleague of mine in the UW Math
department, Neil Koblitz, invented a new approach based on mathematical
objects called ``elliptic curves''; this too involves much number
theory. You literally have an elliptic curve in your cellphone.
Setting aside security, there is still the issue of accuracy. All data
transmission is subject to error. Minor mistakes could erase your bank
account, turn CD's and DVD's into gibberish, and cause a spacecraft to
land in Hoboken instead of on Mars. To prevent this, sophisticated
schemes have been developed that allow the machine receiving the signal
to not only recognize transmission errors, but also correct them in
real-time. These ``error-correcting codes'' are of mathematical origin,
and again involve primes and other aspects of number theory.
As I said at the outset, it's not my goal to talk about applications.
But there are millions of applications out there. If you want to learn
more, try searching ``public-key cryptography'', ``elliptic curve
cryptography'' and ``error-correcting codes''. I can't promise that it
will be easy reading, however.
