The title says it all. My son, who is not a mathematician, pointed it out. It is also, of course, the square of the sum of the first nine numbers.
This last happened in1296 and will not happen again until 3026. Enjoy!
No it does not, because
Nitpick: 1296 is (or was) the sum of the first eight numbers cubed, the sum of the first ten cubes will be 3025, not 3026. Because adding 10^3 = 1,000 is not adding 1,001.
But apart from that, very amusing piece of trivia.
Now I’ll have to check what happened in 1296.
ETA: If this is some kind of numerological foreboding, the new year looks promising:
Tarabya, self-proclaimed king of Pegu, is defeated in single combat on war elephants by Wareru.
I meant 3025 of course.
2025 is also a perfect square (of 45), the first since 1936, and the last until 2116.
(Bolding mine)
Did you write “of course” because it’s trivial, or because there’s an obvious mathematical link between the square of the sum of the first nine numbers, and the sum of the first nine cubes?
Here’s mathematician YouTuber Matt Parker with the same fact, in a slightly different form.
Yes, it is well known (among mathematicians) that the sum of the first cubes is the square of the sum of the first n numbers. E.g., (1+2)^2 = 1 + 8; (1+2+3)^2 = 1+8+27. This fact is not even hard to prove (by induction). The crucial step is showing that
(n+1)^2(n+2)^2 - n^2(n+1)^2 = 4(n+1)
Thanks !
I’m not a mathematician but I love math trivia like this.
That fact is called Nicomachus’s Theorem. The Wikipedia article has more information and several proofs.
2025 is also the smallest number with exactly 15 odd divisors, although that doesn’t seem as interesting as being the sum of the first 9 cubes.
For what it’s worth, it’s something that I didn’t know or (more likely) had forgotten until the past few days when I started seeing memes and references to what this thread is about. It is a cool fact, and (unless I’m missing something) not at all obvious if you haven’t seen it before.
It is more or less trivial that a number has 15 odd divisors it and only if it is some power of 2 times the fourth power of one odd prime times the square of another odd prime. 2025 = 2^03^45^2 is clearly the least such. The next one is 3^4*7^2 = 3969.
Here is another odd fact. AFAIK, it is not part of any pattern:
3^3 + 4^3 + 5^3 = 6^3. This means that 91 = 3^3 + 4^3 = 6^3 + (-5)^3 is the smallest positive integer that is the sum of two integer cubes in two different ways.
And making fun of all the mathing of 2025, I saw this cartoon. Punchline:
1^0+2^0+3^0 + \ldots + 2025^0=2025
And 45 is the sum of the first 9 digits
Well you have square numbers: 44^2 = 1936, 45^2 = 2025, 46^2 = 2116
and you have squares of triangular numbers: 9^210^2/4 = 2025, 10^211^2/4 = 3025
but if you want square triangular numbers then… well after 1225 the next is 41616; due to the constraints the magnitude will grow exponentially.