count

I didn't know where to put this, but then I found /count and knew what I had to do.

Strap in. I catalogued 24 wild and wonderful properties of 2024 so you don't have to: 🧵 👇

@a16zcrypto • Harvard/HBS • 🧩 • QED

2024 = 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 is the 𝘀𝘂𝗺 𝗼𝗳 𝗲𝗶𝗴𝗵𝘁 𝗰𝗼𝗻𝘀𝗲𝗰𝘂𝘁𝗶𝘃𝗲 𝗰𝘂𝗯𝗲𝘀.

Moreover, 2024 is the 𝘀𝘂𝗺 𝗼𝗳 𝗲𝗹𝗲𝘃𝗲𝗻 𝗰𝗼𝗻𝘀𝗲𝗰𝘂𝘁𝗶𝘃𝗲 𝗻𝗼𝗻𝗻𝗲𝗴𝗮𝘁𝗶𝘃𝗲 𝗶𝗻𝘁𝗲𝗴𝗲𝗿𝘀, 𝗯𝘂𝘁 𝗻𝗼 𝗳𝗲𝘄𝗲𝗿 (2024 = 179 + 180 + 181 + 182 + 183 + 184 + 185 + 186 + 187 + 188 + 189).

2024 is a 𝘁𝗲𝘁𝗿𝗮𝗵𝗲𝗱𝗿𝗮𝗹 𝗻𝘂𝗺𝗯𝗲𝗿, obtained by counting the number of balls in successively larger tetrahedral stacks. The most recent such year was 1771, and the next is 2300, followed by 2600!

2024 is also 𝗱𝗼𝗱𝗲𝗰𝗮𝗵𝗲𝗱𝗿𝗮𝗹; the next such is 2925, and then 4060.

There are precisely 2024 𝘄𝗮𝘆𝘀 𝗼𝗳 𝗰𝗵𝗼𝗼𝘀𝗶𝗻𝗴 𝘁𝗵𝗿𝗲𝗲 𝗼𝗯𝗷𝗲𝗰𝘁𝘀 𝗼𝘂𝘁 𝗼𝗳 𝗮 𝘀𝗲𝘁 𝗼𝗳 𝟮𝟰. (This one you might've seen before!)

And there are 2024 𝘀𝗼𝗹𝘂𝘁𝗶𝗼𝗻𝘀 𝘁𝗼 𝘅^𝟮 + 𝘆^𝟮 + 𝘇^𝟮 = 𝟭 𝗺𝗼𝗱𝘂𝗹𝗼 𝟰𝟲.

2024 is the 𝘀𝘂𝗺 𝗼𝗳 𝘁𝗵𝗲 𝗳𝗶𝗿𝘀𝘁 𝟭𝟭 𝗲𝘃𝗲𝗻 𝘀𝗾𝘂𝗮𝗿𝗲𝘀. That's also the 𝗻𝘂𝗺𝗯𝗲𝗿 𝗼𝗳 𝗽𝗼𝘀𝘀𝗶𝗯𝗹𝗲 𝗯𝗶𝘀𝗵𝗼𝗽 𝗺𝗼𝘃𝗲𝘀 𝗼𝗻 𝗮 𝟭𝟮𝘅𝟭𝟮 𝗯𝗼𝗮𝗿𝗱.

And speaking of chess: 2024 is one of the 𝘀𝗾𝘂𝗮𝗿𝗲𝘀 𝘃𝗶𝘀𝗶𝘁𝗲𝗱 𝗯𝘆 𝗮 𝗰𝗵𝗲𝘀𝘀 𝗸𝗶𝗻𝗴 𝗺𝗼𝘃𝗶𝗻𝗴 𝗼𝗻 𝗮 𝘀𝗾𝘂𝗮𝗿𝗲-𝘀𝗽𝗶𝗿𝗮𝗹-𝗻𝘂𝗺𝗯𝗲𝗿𝗲𝗱 𝗯𝗼𝗮𝗿𝗱...

... with the objective of always moving to the adjacent unvisited square containing the number with the most divisors. (Believe it or not, there are only finitely many squares visited in this way – after 1113 steps, the king hits square 855481, and then stops!)

There are 2024 𝗶𝗻𝘁𝗲𝗴𝗲𝗿-𝘀𝗶𝗱𝗲𝗱 𝗾𝘂𝗮𝗱𝗿𝗶𝗹𝗮𝘁𝗲𝗿𝗮𝗹𝘀 𝘄𝗶𝘁𝗵 𝗽𝗲𝗿𝗶𝗺𝗲𝘁𝗲𝗿 𝟰𝟳 (up to rotation)...

... and 2024 𝗶𝗻𝘁𝗲𝗴𝗲𝗿-𝘀𝗶𝗱𝗲𝗱 𝘁𝗿𝗶𝗮𝗻𝗴𝗹𝗲𝘀 𝘄𝗶𝘁𝗵 𝘄𝗶𝘁𝗵 𝗰𝗶𝗿𝗰𝘂𝗺𝗰𝗶𝗿𝗰𝗹𝗲 𝗱𝗶𝗮𝗺𝗲𝘁𝗲𝗿 𝗮𝘁 𝗺𝗼𝘀𝘁 𝟯𝟬...

... and there are 44 integer-sided triangles with perimeter 46, for a total perimeter sum of – you guessed it – 2024.

2024 is the 𝗰𝗶𝗿𝗰𝘂𝗺𝗳𝗲𝗿𝗲𝗻𝗰𝗲 𝗼𝗳 𝘁𝗵𝗲 (𝟰𝟱,𝟰𝟱)-𝗸𝗻𝗶𝗴𝗵𝘁 𝗴𝗿𝗮𝗽𝗵.

2024 is the 𝗻𝘂𝗺𝗯𝗲𝗿 𝗼𝗳 𝗣𝘆𝘁𝗵𝗮𝗴𝗼𝗿𝗲𝗮𝗻 𝘁𝗿𝗶𝗽𝗹𝗲𝘀 𝗺𝗼𝗱𝘂𝗹𝗼 𝟰𝟴.

2024 is the 𝘀𝗺𝗮𝗹𝗹𝗲𝘀𝘁 𝗽𝗼𝘀𝗶𝘁𝗶𝘃𝗲 𝗶𝗻𝘁𝗲𝗴𝗲𝗿 𝘄𝗶𝘁𝗵 𝟭𝟲 𝗱𝗶𝘃𝗶𝘀𝗼𝗿𝘀 𝘀𝘂𝗰𝗵 𝘁𝗵𝗮𝘁 𝘁𝗵𝗲 𝗻𝗲𝘅𝘁 𝗶𝗻𝘁𝗲𝗴𝗲𝗿 𝗵𝗮𝘀 𝟭𝟱 𝗱𝗶𝘃𝗶𝘀𝗼𝗿𝘀.

(The smallest with 17 divisors such that the next integer has 16 divisors is 665416609183179841.)

𝟮𝟬𝟮𝟰^𝟭𝟲 + 𝟮𝟬𝟮𝟰 + 𝟭 =  𝟳𝟵𝟯𝟭𝟳𝟯𝟯𝟴𝟬𝟳𝟴𝟵𝟯𝟯𝟰𝟵𝟯𝟭𝟰𝟵𝟮𝟱𝟯𝟰𝟲𝟵𝟲𝟮𝟬𝟰𝟲𝟱𝟱𝟭𝟭𝟲𝟳𝟬𝟰𝟳𝟴𝟬𝟳𝟮𝟵𝟵𝟴𝟱𝟮𝟵𝟬𝟬𝟭 𝗶𝘀 𝗽𝗿𝗶𝗺𝗲. 👀

It takes 2024 𝗺𝗼𝘃𝗲𝘀 𝗳𝗼𝗿 𝟰𝟰 𝗳𝗿𝗼𝗴𝘀 🐸 𝘁𝗼 𝘀𝘄𝗮𝗽 𝗽𝗹𝗮𝗰𝗲𝘀 𝘄𝗶𝘁𝗵 𝟰𝟰 𝘁𝗼𝗮𝗱𝘀 🐸 𝗼𝗻 𝗮 𝘀𝘁𝗿𝗶𝗽 𝗼𝗳 𝟴𝟵 𝗹𝗶𝗹𝘆 𝗽𝗮𝗱𝘀 (moves are limited to single slides or jumps).

2024 is a 𝗯𝗲𝘁𝗿𝗼𝘁𝗵𝗲𝗱 𝗻𝘂𝗺𝗯𝗲𝗿 (the last such was 1925) and a 𝗝𝗮𝗰𝗸 𝗕𝗮𝘂𝗲𝗿 𝗻𝘂𝗺𝗯𝗲𝗿 (the last such was 1924).

𝟮𝟬𝟮𝟰 = 𝟰𝟲^𝟮 - 𝟮*𝟰𝟲.

Miraculously, (𝟮𝟬𝟮𝟰 + 𝟭)*(𝟮𝟬𝟮𝟰 + 𝟮)*(𝟮𝟬𝟮𝟰 + 𝟯)*(𝟮𝟬𝟮𝟰 + 𝟰)/(𝟮𝟬𝟮𝟰 + (𝟮𝟬𝟮𝟰 + 𝟭) + (𝟮𝟬𝟮𝟰 + 𝟮) + (𝟮𝟬𝟮𝟰 + 𝟯) + (𝟮𝟬𝟮𝟰 + 𝟰)) 𝗶𝘀 𝗮𝗻 𝗶𝗻𝘁𝗲𝗴𝗲𝗿.

The number of integers less than 2024 that are relatively prime to 2024 is equal to the number of integers less than 2024 plus the number of integers less than 2024 that are relatively prime to 2024 that are relatively prime to 2024 plus the number of integers less than 2024 that are relatively prime to 2024.

(This can be expressed more concisely [and slightly less syntactically ambiguously] as 𝗽𝗵𝗶(𝟮𝟬𝟮𝟰) = 𝗽𝗵𝗶(𝟮𝟬𝟮𝟰+𝗽𝗵𝗶(𝟮𝟬𝟮𝟰)), where phi denotes the Euler totient function 😉.)

There are 2024 𝟰𝟲-𝗱𝗶𝗴𝗶𝘁 𝗹𝗲𝗳𝘁- 𝗼𝗿 𝗿𝗶𝗴𝗵𝘁-𝘁𝗿𝘂𝗻𝗰𝗮𝘁𝗮𝗯𝗹𝗲 𝗽𝗿𝗶𝗺𝗲𝘀. 

There exists only one such 60-digit prime, namely 202075909708030901050930450609080660821035604908735717137397.

ICYMI: this New Year's Day is like none other this century...

... because the date, 1/1/24, lines up with the first four nonzero Tribonacci numbers! 🥳🥳🥳🥳

(Like Fibonacci numbers, except that you add every *three* numbers to get the next one.)

Commemorative OE: https://app.manifold.xyz/c/Numerology-1124

Oh also: https://warpcast.com/skominers/0x6074816a

Plus 2024 (= 44*46) is the first year since 888 that both has 𝗮𝗹𝗹 𝗲𝘃𝗲𝗻 𝗱𝗶𝗴𝗶𝘁𝘀 𝗮𝗻𝗱 𝗰𝗮𝗻 𝗯𝗲 𝗳𝗮𝗰𝘁𝗼𝗿𝗲𝗱 𝗶𝗻𝘁𝗼 𝘁𝘄𝗼 𝗻𝘂𝗺𝗯𝗲𝗿𝘀 𝘄𝗶𝘁𝗵 𝗮𝗹𝗹 𝗲𝘃𝗲𝗻 𝗱𝗶𝗴𝗶𝘁𝘀.