Matthew van Eerde's web log

If you've bowled, you know the arrangement of the bowling pins forms a triangle.

(Image courtesy of the International Pumpkin Federation.)

If you've played eight-ball, you know the arrangement of the fifteen billiard balls forms a triangle.

Ten, fifteen... what other numbers form a triangle? The common arrangement of nine-ball and ninepins doesn't count because it's a diamond, not a triangle.

You can start with ten and add five balls to make a triangle of fifteen... then add six more to make a triangle of 21... then seven more to make a triangle of 28... and so on, with this sequence:

..., 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231...

Start looking for patterns.  What do you see?  Nothing jumps out right away.  Are there any primes?  Ones that end in 0, 2, 4, 6, or 8 are obviously even and therefore not primes... ones that end in 5 are divisible by 5 and therefore not primes... the remaining ones fall due to specific cases (21 = 3 x 7; 91 = 7 x 13; 153 = 3 x 3 x 17; 171 = 3 x 3 x 19; 231 = 3 x 7 x 11...)

In fact, gosh darn it, it seems like none of these numbers are prime, no matter how far we extend that "..." on the right!  Can this be a coincidence?

A few mental coruscations later I have an idea that it is not a coincidence... that triangular numbers, by there very nature, cannot be prime.  In fact I'm even willing to call it a "proof."  Here it is.

"Theorem": there are no triangular primes.

First we need to generate a formula for the triangular numbers.  Note that if you take an n x (n + 1) rectangle and draw a zig-zag line like so, you get two triangles.

Each of these triangles is composed of n diagonals, the shortest of which is 1, and the longest of which is n.  That is to say, each of the triangles is composed of t_n squares.  So we know that the total number of squares in the rectangle is 2t_n.

But we also know that the total number of squares is the base times the height, or n(n + 1).  This gives us 2t_n = n(n + 1), or t_n = n(n + 1)/2.

The next part of the "proof" breaks down into case analysis.  n can be odd (as in the diagram, where n is 5) or even.

Case where n is even:

n is an even positive number.  Therefore n/2 is a positive number (maybe odd, maybe even; doesn't matter.)  t_n can be written as (n/2)(n + 1), and is therefore not prime, since it has at least four factors: 1, n/2, n + 1, and t_n.

Case where n is odd:

n is an odd positive number.  Therefore n + 1 is an even positive number.  Therefore (n + 1)/2 is a positive number (maybe odd, maybe even; doesn't matter.)  t_n can be written as (n)([n + 1]/2), and is therefore not prime, since it has at least four factors: 1, n, (n + 1)/2, and t_n.

Note that the "proof" that t_n is not prime inevitably concludes in a stronger result - that t_n has at least four factors... not only is t_n not prime, it can't even have as few as three factors.  (Some numbers with three factors: 4, 9, 25, 49...)

Exercise: what kind of numbers have exactly three factors?

A beautiful proof.  Perhaps the two cases can be elegantly folded together to normalize it a bit better.  That is not a serious problem.

There is a serious problem.

The proof of our result is doomed.

Why?

Because the result does not hold! There is a triangular prime.

Exercise: find a triangular prime.

After having recovered from the shocking revelation that, our beautiful proof to the contrary, a triangular prime is so rude as to exist, a little self-examination is in order.  What is wrong with the proof?  This... and not the existence of triangular primes... is the lesson to be learned: that beauty is not always truth.