S
, S,
S
, P,
P
, P,
S
, S,
P
, P,
S
, S,
S
, P,
S
, P,
P
, . . .
Is there some kind of pattern here, or is there none? What do you think? If we pull out just the Class-A letters, we get this:
SSPSPSSSP
; and if we pull out just the Class-B letters, we get this: SPPSPSPP. If there is any kind of periodicity or subtler type of rhythmicity here, it’s certainly elusive. No simple predictable pattern jumps out either in boldface or in non-boldface, nor did any jump out when they were mixed together. We have picked up a hint of a quite even balance between the two classes, yet we lack any hints as to why that might be. This is provocative but frustrating.
People who Pursue Patterns with Perseverance
At this juncture, I feel compelled to point out a distinction not between two classes of numbers, but between two classes of people. There are those who will immediately be drawn to the idea of pattern-seeking, and there are those who will find it of no appeal, perhaps even distasteful. The former are, in essence, those who are mathematically inclined, and the latter are those who are not. Mathematicians are people who at their deepest core are drawn on — indeed, are easily seduced — by the urge to find patterns where initially there would seem to be none. The passionate quest after order in an apparent disorder is what lights their fires and fires their souls. I hope you are among this class of people, dear reader, but even if you are not, please do bear with me for a moment.
It may seem that we have already divined a pattern of sorts — namely, that we will forever encounter just singletons and pairs. Even if we can’t quite say how the S’s and P’s will be interspersed, it appears at least that the imposition of the curious dichotomy “sums-of-two-squares
vs.
not-sums-of-two-squares” onto the sequence of the prime numbers breaks it up into singletons and pairs, which is already quite a fantastic discovery! Who would have guessed?
Unfortunately, I must now confess that I have misled you. If we simply throw the very next prime, which is 101, into our list, it sabotages the seeming order we’ve found. After all, the prime number 101, being the sum of the two squares 1 and 100, and thus belonging to Class A, has to be written in boldface, and so our alleged boldface
pair
89–97 turns out to be a boldface
triplet
instead. And thus our hopeful notion of a sequence of just S’s and P’s goes down the drain.
What does a pattern-seeker do at this point — give up? Of course not! After a setback, a flexible pattern-seeker merely
regroups.
Indeed, taking our cue from the word just given, let us try regrouping our sequence of primes in a different fashion. Suppose we segregate the two classes, displaying them on separate lines. This will give us the following:
Yes
square + square: 2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101,…
No
square + square: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83,…
Do you see anything yet? If not, let me give you a hint. What if you simply take the differences between adjacent numbers in each line? Try it yourself — or else, if you’re very lazy, then just read on.
In the upper line, you will get 3, 8, 4, 12, 8, 4, 12, 8, 12, 16, 8, 4, whereas in the lower line you will get 4, 4, 8, 4, 8, 12, 4, 12, 8, 4, 8, 4. There is something that surely should jump out at even the most indifferent reader at this point: not only is there a preponderance of just a few integers (4, 8, and 12), but moreover, all these integers are multiples of 4. This seems too much to be merely coincidental.
And the only larger number in these two lists — 16 — is also a multiple of 4. Will this new pattern — multiples of 4 exclusively — hold up forever? (Of course, there is that party-pooper of a ‘3’ at the very outset, but we can chalk it up to the fact that 2 is the only even prime. No big deal.)
Where There’s Pattern, There’s Reason
The key thought in the preceding few lines is the article of faith that
this pattern cannot merely be a coincidence.
A mathematician who finds a pattern of this sort will instinctively ask, “Why? What is the
reason
behind this order?” Not only will all mathematicians wonder
what
the reason is, but even more importantly, they will all implicitly believe that whether or not anyone ever finds the reason, there
must be
a reason for it. Nothing happens “by accident” in the world of mathematics. The existence of a perfect pattern, a regularity that goes on forever, reveals — just as smoke reveals a fire — that something is going on behind the scenes. Mathematicians consider it a sacred goal to seek that thing, uncover it, and bring it out into the open.
This activity is called, as you well know, “finding a proof ”, or stated otherwise, turning a conjecture into a theorem. The late great eccentric Hungarian mathematician Paul Erdös once made the droll remark that “a mathematician is a device for turning coffee into theorems”, and although there is surely truth in his witticism, it would be more accurate to say that mathematicians are
devices for finding conjectures and turning them into theorems.
What underlies the mathematical mindset is an unshakable belief that whenever some mathematical statement X is
true,
then X has a
proof,
and vice versa. Indeed, to the mathematical mind, “having a proof ” is no more and no less than what “being true” means! Symmetrically, “being false” means “having no proof ”. One can find
hints
of a perfect, infinite pattern by doing numerical explorations, as we did above, but how can one know for sure that a suspected regularity will continue forever, without end? How can one know, for instance, that there are infinitely many prime numbers? How do we know there will not, at some point, be a last one — the Great Last Prime
P
?
If it existed,
P
would be a truly important and interesting number, but if you look at a long list of consecutive primes (the list above of primes up to 100 gives the flavor), you will see that although their rhythm is a bit “bumpy”, with odd gaps here and there, the interprime gaps are always quite small compared to the size of the primes involved. Given this very clear trend, if the primes were to run out all of a sudden, it would almost feel like falling off the edge of the Earth without any warning. It would be a huge shock. Still, how do we
know
this won’t happen? Or do we know it? Finding, with the help of a computer, that new primes keep on showing up way out into the billions and the trillions is great, but it won’t guarantee in rock-solid fashion that they won’t just stop all of a sudden somewhere out further. We have to rely on
reasoning
to get us there, because although finite amounts of evidence can be strongly suggestive, they just don’t cut the mustard, because infinity is very different from any finite number.
Sailing the Ocean of Primes and Falling off the Edge
You probably have seen Euclid’s proof of the infinitude of the primes somewhere, but if not, you have missed out on one of the most crucial pillars of human knowledge that ever have been found. It would be a gap in your experience of life as sad as never having tasted chocolate or never having heard a piece of music. I can’t tolerate such crucial gaps in my readers’ knowledge, so here goes nothing!
Let’s suppose that
P,
the Great Last Prime in the Sky, does exist, and see what that supposition leads to. For
P
to exist means that there is a Finite, Closed Club of All Primes, of which
P
itself is the glorious, crowning, final member. Well then, let’s boldly multiply all the primes in the Closed Club together to make a delightfully huge number called
Q.
This number
Q
is thus divisible by 2 and also by 3, 5, 7, 11, and so forth. By its definition,
Q
is divisible by every prime in the Club, which means by every prime in the universe! And now, for a joyous last touch, as in birthday parties, let’s add one candle to grow on, to make
Q
+ 1. So here’s a colossal number that, we are assured, is
not
prime, since
P
(which is obviously dwarfed by
Q
) is the Great Last Prime, the biggest prime of all. All numbers beyond
P
are, by our initial supposition, composite. Therefore
Q
+ 1, being way beyond
P
and hence composite,
has
to have some prime divisor. (Remember this, please.)
What could that unknown prime divisor be? It can’t be 2, because 2 divides
Q
itself, which is just one step below
Q
+ 1, and two even numbers are never located at a distance of 1 from each other. It also can’t be 3, because 3 likewise divides
Q
itself, and numbers divisible by 3 are never next-door neighbors! In fact, whatever prime
p
that we select from the Club, we find that
p
can’t divide
Q
+ 1, because
p
divides its lower neighbor
Q
(and multiples of
p
are never next-door neighbors — they come along only once every
p
numbers). And so reasoning has shown us that
none
of the members of the Finite, Closed Club of Primes divides
Q
+ 1.
But above, I observed (and I asked you to remember) that
Q
+ 1, being composite,
has
to have a prime divisor. Sting! We have been caught in a trap, painted ourselves into a corner. We have concocted a crazy number — a number that on the one hand must be composite (
i.e.,
has some smaller prime divisor) and yet on the other hand has no smaller prime divisor. This contradiction came out of our assumption that there was a Finite, Closed Club of Primes, gloriously crowned by
P
, and so we have no choice but to go back and erase that whole amusing, suspect vision.
There cannot be a “Great Last Prime in the Sky”; there cannot be a “Finite, Closed Club of All Primes”. These are fictions. The truth, as we have just demonstrated, is that the list of primes goes on without end. We will never, ever “fall off the Earth”, no matter how far out we go. Of that we now are assured by flawless reasoning, in a way that no
finite
amount of computational sailing among seas of numbers could ever have assured us.
If, perchance, coming to understand
why
there is no last prime (as opposed to merely knowing
that
it is the case) was a new experience to you, I hope you savored it as much as a piece of chocolate or of music. And just like such experiences, following this proof is a source of pleasure that one can come back to and dip into many times, finding it refreshing each new time. Moreover, this proof is a rich source of other proofs — Variations on a Theme by Euclid (though we will not explore them here).
The Mathematician’s Credo
We have just seen up close a lovely example of what I call the “Mathematician’s Credo”, which I will summarize as follows:
X is true
because
there is a proof of X;
X is true
and so
there is a proof of X.
Notice that this is a two-way street. The first half of the Credo asserts that
proofs are guarantors of truth,
and the second half asserts that
where there is a regularity, there is a reason.
Of course we ourselves may not uncover the hidden reason, but we firmly and unquestioningly believe that it exists and in principle could someday be found by someone.
To doubt either half of the Credo would be unthinkable to a mathematician. To doubt the first line would be to imagine that a proved statement could nonetheless be false, which would make a mockery of the notion of “proof ”, while to doubt the second line would be to imagine that within mathematics there could be perfect, exceptionless patterns that go on forever, yet that do so with no rhyme or reason. To mathematicians, this idea of flawless but reasonless structure makes no sense at all. In that regard, mathematicians are all cousins of Albert Einstein, who famously declared, “God does not play dice.” What Einstein meant is that nothing in nature happens without a cause, and for mathematicians, that there is always one unifying, underlying cause is an unshakable article of faith.
No Such Thing as an Infinite Coincidence
We now return to Class A versus Class B primes, because we had not quite reached our revelation, had not yet experienced that mystical
frisson
I spoke of. To refresh your memory, we had noticed that each line was characterized by differences of the form 4
n
— that is, 4, 8, 12, and so forth. We didn’t
prove
this fact, but we
observed
it often enough that we
conjectured
it.
The lower line in our display starts out with 3, so our conjecture would imply that all the other numbers in that line are gotten by adding various multiples of 4 to 3, and consequently, that every number in that line is of the form 4
n
+ 3. Likewise (if we ignore the initial misfit of 2), the first number in the upper line is 5, so if our conjecture is true, then every subsequent number in that line is of the form 4
n
+ 1.
Well, well — our conjecture has suggested a remarkably simple pattern to us: Primes of the form 4
n
+ 1
can
be represented as sums of two squares, while primes of the form 4
n
+ 3
cannot.
If this guess is correct, it establishes a beautiful, spectacular link between primes and squares (two classes of numbers that
a priori
would seem to have nothing to do with each other), one that catches us completely off guard. This is a glimpse of pure magic — the kind of magic that mathematicians live for.