What are the interesting ways to write $2024$ and some facts about $2024$?
We know that $2024$ is coming in $48$ hours, so I decided to ask this.
It is well known fact that $45^2=2025$, so $2024$ can be written as product of two consecutive even numbers, namely $44$ and $46$. The next year that is one less than a perfect is $2115$(about $9$ decades from now), and the next year that is one less tahn a cube is $2196$(about $17$ decades away from now).
$2024$ is a tetrahedral number, according to Wikipedia.
Another than above, what are some more special facts about $2024$?
Here are some interesting ways to write $2024$:
$2^{3!+2!^{2}+!2}-4!=2024$(this is the only solution to $x^{11}-y!=2024$)
$2^{10}+10^3=2024$
$(3!\cdot2!+2!)^3-(2+1)!=2024$
$45^2-1!=2024$
$!7+5!+4!+4!+1!=1024$
$2024=!5\cdot(!5+2$)
$F_{15}\equiv2024\pmod{11^4}$
$2^{10}+0+2^{10}-4!=2024$
In base thirty-four, $2024$ is written as $1pi_{34}$.
In base fourty-four(also repdigit), $2024$ is repdigit because it can be written as $(44:44)_{44}$. Although this is somehow trival.
What are more interesting ways to write $2024$?
Prime factorization
$$2024 = 2^3 \times 11 \times 23$$
Divisors of $2024$
$$\{1,2,4,8,11,22,23,44,46,88,92,184,253,506,1012,2024\}$$
2024 is a Harshad Number See Harshad number
The $2024$th prime $$17599$$
Factor $2024$ using Gaussian integers $2024 = i\times (1+i)^6\times 11 \times 23$
The exoplanets that have a mass that is roughly $2024$ times that of the Earth
From https://oeis.org/A002492, $2024$ is equal to the Sum of the first $n$ even squares: $\frac{2n(n+1)(2n+1)}{3}$ for $n = 11$, one of its factors
$$\{0,4,20,56,120,220,364,560,816,1140,1540,2024\}$$
From https://oeis.org/A000292, $2024$ is one of the Tetrahedral (or triangular pyramidal) numbers, $a(n) = C(n+2,3) = \frac{n(n+1)(n+2)}{6}$, for $n=22$
$$\{0,1,4,10,20,35,56,84,120,165,220,286,364,455,560,680,816,969,1140,1330,1540,1771,2024\}$$
$2024$ is found in $491$ OEIS results from $367,000$ sequences, per https://oeis.org/search?q=2024&go=Search as of $2024$
Write $2024$ as the sum of Fibonacci numbers
$$1597+377+34+13+3\\1597+377+34+13+2+1\\1597+377+34+8+5+3\\ 1597+233+144+34+13+3\\ 987+610+377+34+13+3\\ 1597+377+34+8+5+2+1\\ 1597+377+21+13+8+5+3\\ 1597+233+144+34+13+2+1\\ 1597+233+144+34+8+5+3\\ 1597+233+89+55+34+13+3\\ 987+610+377+34+13+2+1\\ 987+610+377+34+8+5+3\\ 987+610+233+144+34+13+3\\ 1597+377+21+13+8+5+2+1\\ 1597+233+144+34+8+5+2+1\\ 1597+233+144+21+13+8+5+3\\ 1597+233+89+55+34+13+2+1\\ 1597+233+89+55+34+8+5+3\\ 987+610+377+34+8+5+2+1\\ 987+610+377+21+13+8+5+3\\ 987+610+233+144+34+13+2+1\\ 987+610+233+144+34+8+5+3\\ 987+610+233+89+55+34+13+3\\ 1597+233+144+21+13+8+5+2+1\\ 1597+233+89+55+34+8+5+2+1\\ 1597+233+89+55+21+13+8+5+3\\ 987+610+377+21+13+8+5+2+1\\ 987+610+233+144+34+8+5+2+1\\ 987+610+233+144+21+13+8+5+3\\ 987+610+233+89+55+34+13+2+1\\ 987+610+233+89+55+34+8+5+3\\ 1597+233+89+55+21+13+8+5+2+1\\ 987+610+233+144+21+13+8+5+2+1\\ 987+610+233+89+55+34+8+5+2+1\\ 987+610+233+89+55+21+13+8+5+3\\ 987+610+233+89+55+21+13+8+5+2+1$$
Find a random monic polynomial that has $2024$ as a root
$$x^4-2019 x^3-10119 x^2-2019 x-10120=0$$
Write $2024$ as the sum of three squares, for example, $2024= 2^2 + 16^2 + 42^2$
$$\begin{array}{ccc} 2 & 16 & 42 \\ 2 & 24 & 38 \\ 8 & 14 & 42 \\ 10 & 18 & 40 \\ 10 & 30 & 32 \\ 16 & 18 & 38 \\ 18 & 26 & 32 \\ \end{array}$$
Show Goldbach Conjecture that every even counting number greater than 2 is equal to the sum of two prime numbers, that is $2024 = a + b$
$$\{a\to 73,b\to 1951\}\\\{a\to 157,b\to 1867\}\\\{a\to 577,b\to 1447\}\\\{a\to 811,b\to 1213\}\\\{a\to 991,b\to 1033\}\\\{a\to 1117,b\to 907\}\\\{a\to 1423,b\to 601\}\\\{a\to 1627,b\to 397\}\\\{a\to 1747,b\to 277\}\\\{a\to 1783,b\to 241\}\}$$
Find non-negative solutions to $20 x_1 +24 x_2 =2024$
$$\begin{array}{cc} 4 & 81 \\ 10 & 76 \\ 16 & 71 \\ 22 & 66 \\ 28 & 61 \\ 34 & 56 \\ 40 & 51 \\ 46 & 46 \\ 52 & 41 \\ 58 & 36 \\ 64 & 31 \\ 70 & 26 \\ 76 & 21 \\ 82 & 16 \\ 88 & 11 \\ 94 & 6 \\ 100 & 1 \\ \end{array}$$
Image of the output of a cellular automaton with rule $110$ where the initial condition is the binary representation of $2024$
Draw a word cloud from US holidays from https://www.timeanddate.com/holidays/us/2024
Write $2024 = a + b + c$, where $a, b, c$ are primes
$$\{\{a\to 1601,b\to 2,c\to 421\},\{a\to 1873,b\to 149,c\to 2\},\{a\to 1009,b\to 2,c\to 1013\},\{a\to 1453,b\to 569,c\to 2\},\{a\to 2,b\to 739,c\to 1283\},\{a\to 2,b\to 1499,c\to 523\},\{a\to 2,b\to 439,c\to 1583\},\{a\to 239,b\to 1783,c\to 2\},\{a\to 2,b\to 1321,c\to 701\},\{a\to 2,b\to 1741,c\to 281\},\{a\to 2,b\to 1543,c\to 479\},\{a\to 1663,b\to 2,c\to 359\},\{a\to 2,b\to 1831,c\to 191\},\{a\to 401,b\to 1621,c\to 2\},\{a\to 1669,b\to 2,c\to 353\},\{a\to 149,b\to 1873,c\to 2\},\{a\to 1933,b\to 89,c\to 2\},\{a\to 1613,b\to 409,c\to 2\},\{a\to 2011,b\to 11,c\to 2\},\{a\to 2,b\to 919,c\to 1103\},\{a\to 191,b\to 2,c\to 1831\},\{a\to 73,b\to 2,c\to 1949\},\{a\to 1459,b\to 563,c\to 2\},\{a\to 1951,b\to 2,c\to 71\},\{a\to 701,b\to 2,c\to 1321\},\{a\to 2,b\to 571,c\to 1451\},\{a\to 2,b\to 2003,c\to 19\},\{a\to 821,b\to 1201,c\to 2\},\{a\to 2011,b\to 2,c\to 11\},\{a\to 29,b\to 2,c\to 1993\},\{a\to 2,b\to 1949,c\to 73\},\{a\to 821,b\to 2,c\to 1201\},\{a\to 919,b\to 1103,c\to 2\},\{a\to 1951,b\to 71,c\to 2\},\{a\to 491,b\to 2,c\to 1531\},\{a\to 1709,b\to 2,c\to 313\},\{a\to 2,b\to 773,c\to 1249\},\{a\to 1249,b\to 2,c\to 773\},\{a\to 599,b\to 2,c\to 1423\},\{a\to 1753,b\to 269,c\to 2\},\{a\to 1979,b\to 2,c\to 43\},\{a\to 2,b\to 733,c\to 1289\},\{a\to 2,b\to 421,c\to 1601\},\{a\to 1741,b\to 281,c\to 2\},\{a\to 2,b\to 191,c\to 1831\},\{a\to 313,b\to 1709,c\to 2\},\{a\to 1669,b\to 353,c\to 2\},\{a\to 2,b\to 1039,c\to 983\},\{a\to 1933,b\to 2,c\to 89\},\{a\to 1783,b\to 2,c\to 239\},\{a\to 463,b\to 2,c\to 1559\},\{a\to 421,b\to 2,c\to 1601\},\{a\to 1523,b\to 2,c\to 499\},\{a\to 191,b\to 1831,c\to 2\},\{a\to 2,b\to 661,c\to 1361\},\{a\to 2,b\to 719,c\to 1303\},\{a\to 1913,b\to 2,c\to 109\},\{a\to 2,b\to 1279,c\to 743\},\{a\to 2,b\to 19,c\to 2003\},\{a\to 1013,b\to 1009,c\to 2\},\{a\to 2,b\to 599,c\to 1423\},\{a\to 1709,b\to 313,c\to 2\},\{a\to 353,b\to 2,c\to 1669\},\{a\to 89,b\to 2,c\to 1933\},\{a\to 1381,b\to 641,c\to 2\},\{a\to 149,b\to 2,c\to 1873\},\{a\to 313,b\to 2,c\to 1709\},\{a\to 2017,b\to 5,c\to 2\},\{a\to 199,b\to 1823,c\to 2\},\{a\to 5,b\to 2017,c\to 2\},\{a\to 2,b\to 1979,c\to 43\},\{a\to 1789,b\to 233,c\to 2\},\{a\to 409,b\to 2,c\to 1613\},\{a\to 929,b\to 2,c\to 1093\},\{a\to 571,b\to 2,c\to 1451\},\{a\to 2,b\to 409,c\to 1613\},\{a\to 2,b\to 71,c\to 1951\},\{a\to 2,b\to 613,c\to 1409\},\{a\to 1543,b\to 479,c\to 2\},\{a\to 11,b\to 2011,c\to 2\},\{a\to 2,b\to 313,c\to 1709\},\{a\to 2,b\to 353,c\to 1669\},\{a\to 73,b\to 1949,c\to 2\},\{a\to 2,b\to 443,c\to 1579\},\{a\to 2,b\to 983,c\to 1039\},\{a\to 1103,b\to 2,c\to 919\},\{a\to 1499,b\to 2,c\to 523\},\{a\to 2,b\to 199,c\to 1823\},\{a\to 1069,b\to 2,c\to 953\},\{a\to 281,b\to 2,c\to 1741\},\{a\to 2,b\to 1663,c\to 359\},\{a\to 1163,b\to 859,c\to 2\},\{a\to 263,b\to 2,c\to 1759\},\{a\to 269,b\to 2,c\to 1753\},\{a\to 2,b\to 1913,c\to 109\},\{a\to 1543,b\to 2,c\to 479\},\{a\to 1949,b\to 73,c\to 2\},\{a\to 809,b\to 1213,c\to 2\},\{a\to 43,b\to 2,c\to 1979\},\{a\to 1979,b\to 43,c\to 2\}\}$$
Other interesting facts about 2024 can be found at https://www.numbersaplenty.com/2024
$2024$ in Roman Numerals is $MMXXIV$