I was invited to attend Gathering 4 Gardner this year
(FINALLY!) and had a blast. Every single person I met was utterly fascinating
in their own right – I met an unlikely number of interesting people! And, I was
lucky enough to be able to give a talk at Gathering 4 Gardner, which was (to my
surprise) well-received. I was afraid my half-baked musings would be too simple
for that caliber of mathematician, but it turns out that they place a lot of
value on something being understandable for someone of all skill levels. So
here’s a brief discussion of the talk I gave!
A prime number is a number whose only divisors are 1 and
itself. I’ve been thinking about different ways to define prime numbers, and
here’s another definition that is completely legitimate: A prime number is the
area of an integer-sided rectangle whose length or width must be 1. In the
picture below, we can see that 12 is not a prime, because it can be written as
a 6x2 or a 4x3 rectangle. 13 is a prime, because it can only be in an 13x1
rectangle.
And voila! Our new definition gives us something to
manipulate and explore. What happens if we change “rectangle” to “triangle” or
“hexagon”? What kinds of things are triprimes, or hexprimes?
A few considerations about technique and assumptions first.
The defining characteristic of a rectangle as opposed to other quadrilaterals
is that rectangles are equiangular, so I kept the triangles and hexagons
equiangular as well. Rectangles are also very easy to break down the area of –
triangles and especially hexagons are not quite as easy to break down. So I use
a circle-packing metric in the place of area: for squares I use circles packed
in a square grid, and for triangles and hexagons I use an isometric grid.
So, triprimes turn out to be kinda boring. Equiangular
triangles are equilateral, so the only numbers that can be expressed as a
triangle at all are, of course, the triangle numbers (1,3,6,10,15,21,…). The
concept of the triangle numbers literally arose from this fact. So there’s a
bunch of very predictable tricomposites (composites because their expressions
do not have sides of length 1) and not really any triprimes. Like I said,
boring.
Hexprimes, on the other hand, are fascinating. The
equiangular constraint leaves plenty of wiggle room for interesting hexagons to
form for each possible area. Some of those hexagons are diagrammed below. Note,
they all have at least one tiny side, meaning they’re all prime.
And then we get to 7, the first hexcomposite. This hexagon
with area 7 has no sides of length 1, which means it’s a hexcomposite. No
smaller number is a hexcomposite!
Now that we know that both hexprimes and hexcomposites
exist, it’s only natural to ask what other numbers are hexprimes and
hexcomposites. It turns out that finding new hexcomposites is a bit different
than finding regular composites. With rectangles, you can just double a
rectangle and put them together to create another rectangle that must be a
composite. You can’t combine an arbitrary hexagon with itself to create another
hexagon. No, you have to increase by rows, like shown below.
And it turns out you can do this in a lot of different ways.
So, expressions for hexcomposites converge very quickly.
With these hexagons, I can show that any number > 17 must be a
hexcomposite! And that means that there are a finite number of hexprimes, and
we can find them easily by checking the cases below 18. And, drum roll please –
there are exactly 10 hexprimes! 2, 3, 4, 5, 6, 8, 9, 11, 15, and 17!
A final note – triprimes are very restrictive, and
imminently predictable. Hexprimes are less restrictive, but composite-finding
techniques converge quickly, so they are predictable as well. Give me any
positive number, and I can very quickly tell you if it’s a triprime or a
hexprime. But the middle ground, “rectangular” primes, AKA regular old prime
numbers, are still infrequent and infinite enough to be unpredictable!
