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 2

*t*.

_{n}But we also know that the total number of squares is the base times the height, or *n*(*n* + 1). This gives us 2*t _{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*has at least

_{n}*four*factors... not only is

*t*not prime, it can't even have as few as

_{n}*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.