#3003

First, note that 3,003 is a triangular number. Triangular numbers count objects that can be arranged into an equilateral triangle, as such:

Hopefully you can see that you can generate triangular numbers by adding up the natural numbers to a specific n. Triangular numbers therefore have the form: T_n = 1 + 2 + 3 + . . . + n

You may remember that summing up the numbers like this has the closed form

T_n = n(n + 1) / 2

If we let n = 77, then 77 (78) / 2 = 3,003

That means 3,003 objects can be arranged perfectly into a triangle with 77 objects on each side.

However, there is something even more interesting about this number, namely that it is related to Pascal's triangle, which is constructed as follows: In the top row, let there be a unique nonzero entry 1. Then each subsequent row can be made by adding the number above and to the left to the number above and to the right, treating blank entries as 0.

Equivalently, the entries of Pascal’s triangle are binomial coefficients. The kth entry in the nth row is denoted (n, k), pronounced "n choose k," which counts the number of ways to choose k objects from n.

Note that 3,003 = (14, 6), but because of symmetry, (14, 6) = (14, 8). So 3,003 appears at least twice in this table. However, that's not too unusual. Many numbers appear twice due to symmetry.

What makes 3,003 special is that it is the only known number (other than 1) that appears 8 times in Pascal's triangle. No other number (except 1, which appears infinitely many times along the edges) appears more often. Most numbers appear once or twice, and occasionally 4 times, but appearing 8 times is incredibly rare.

This phenomenon is tied to a famous unsolved problem called Singmaster's conjecture, which states that any natural number greater than 1 appears only finitely many times in Pascal's triangle. Even more interestingly, it is conjectured that there is a universal upper bound.

Finally, just as a quick note about 3,003's structure, it is factored as 3 x 7 x 11 x 13, which is a beautiful product of four consecutive primes (skipping 5). It also equals (15, 5), and several other binomial identities produce it as well, which is why it repeats so often.

Good dog.

Luckily for everyone, these Underlings still can't stand up to a true First Guardian.

Bec's effortless victory here is a very good sign. If this fight was anything close to a challenge for him, then he wouldn't have a hope against Jack.

You finally hop off the lowly GREENTIKE rung and secure your position on the somewhat respectable KIDDO ECLIPSE rung.

Anyway - welcome to Sburb, Jade Harley.

Your Entry might have been a mess, but it was still one of the biggest milestones so far. Just for a moment, let's take a breather, and celebrate our fully populated Medium!

It's finally time to hear from a First Guardian who's on our side.

Whatever else Bec is, he's still a dog. I expect him to have speech patterns comparable to Jaspers, but with the ominous presence and knowledge of Doc Scratch. It'll be an interesting combination.

Oh, for the love of peace.

...actually, let me try a couple things. I haven't had a chance to datamine Homestuck before, and I want to see if there's anything hidden in these gifs.

...gif, singular. All eight of these images link to the same file.

Well, I checked out the metadata, and even plugged it into Hexedit, but I couldn't find anything meaningful - except that Bec is a fan of Netscape. Gotta loop those gifs somehow!

You have extraordinarily bad timing. Her guardian will not be pleased with your intrusion.

Another prophecy fulfilled.