# 2025: sum of the first nine cubes **URL:** \ **Category:** Miscellaneous and Personal Stuff I Must Share\ **Tags:** science-math\ **Created:** [January 1, 2025, 4:37pm UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443 "2025-01-01T16:37:42Z")\ **Posts on this page:** 14\ **Page:** 1
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [January 1, 2025, 4:37pm UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443/1 "2025-01-01T16:37:42Z")
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! ---
**Author:** ![Pardel-Lux](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pardel-lux/32/20182_2.png) [@Pardel-Lux](https://boards.straightdope.com/u/Pardel-Lux)\ **Post date:** [January 1, 2025, 4:50pm UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443/2 "2025-01-01T16:50:19Z")
> [@Hari\_Seldon](#): > > The title says it all. 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](https://en.wikipedia.org/wiki/Tarabya_of_Pegu), self-proclaimed [king of Pegu](https://en.wikipedia.org/wiki/List_of_rulers_of_Pegu), is defeated in single combat on war elephants by [Wareru](https://en.wikipedia.org/wiki/Wareru). ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [January 1, 2025, 9:17pm UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443/3 "2025-01-01T21:17:59Z")
I meant 3025 of course. ---
**Author:** ![commasense](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/commasense/32/3017_2.png) [@commasense](https://boards.straightdope.com/u/commasense)\ **Post date:** [January 2, 2025, 3:55am UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443/4 "2025-01-02T03:55:19Z")
2025 is also a perfect square (of 45), the first since 1936, and the last until 2116. ---
**Author:** ![Moonrise](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/moonrise/32/14764_2.png) [@Moonrise](https://boards.straightdope.com/u/Moonrise)\ **Post date:** [January 2, 2025, 9:51am UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443/5 "2025-01-02T09:51:33Z")
> [@Hari\_Seldon](#): > > 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. (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? ---
**Author:** ![GuanoLad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/guanolad/32/42_2.png) [@GuanoLad](https://boards.straightdope.com/u/GuanoLad)\ **Post date:** [January 2, 2025, 10:43am UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443/6 "2025-01-02T10:43:06Z")
Here’s mathematician YouTuber Matt Parker with the same fact, in a slightly different form. > **[Matt Parker (@standupmaths.bsky.social)](https://bsky.app/profile/standupmaths.bsky.social/post/3lem5uzr5lc24)** > > Because 2025 = 45² and 45 is a triangle number, we get these lovely sums: > > 2025 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)² > 2025 = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ > > Pleasingly the next triangle number 55 is 10 more, so get ready for a repeat of... ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [January 2, 2025, 3:14pm UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443/7 "2025-01-02T15:14:54Z")
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) ---
**Author:** ![Moonrise](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/moonrise/32/14764_2.png) [@Moonrise](https://boards.straightdope.com/u/Moonrise)\ **Post date:** [January 2, 2025, 4:17pm UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443/8 "2025-01-02T16:17:12Z")
Thanks ! I’m not a mathematician but I love math trivia like this. ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [January 2, 2025, 6:48pm UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443/9 "2025-01-02T18:48:23Z")
That fact is called Nicomachus’s Theorem. The [Wikipedia article](https://en.wikipedia.org/wiki/Squared_triangular_number) 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. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 2, 2025, 7:21pm UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443/10 "2025-01-02T19:21:57Z")
> [@Hari\_Seldon](#): > > 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. 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. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [January 2, 2025, 8:32pm UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443/11 "2025-01-02T20:32:53Z")
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 = 2^0_3^4_5^2 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. ---
**Author:** ![echoreply](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/echoreply/32/3641_2.png) [@echoreply](https://boards.straightdope.com/u/echoreply)\ **Post date:** [January 2, 2025, 9:29pm UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443/12 "2025-01-02T21:29:55Z")
And making fun of all the mathing of 2025, I saw this cartoon. Punchline: 1^0+2^0+3^0 + \ldots + 2025^0=2025 > **[cts (@gf256.bsky.social)](https://bsky.app/profile/gf256.bsky.social/post/3lervohy7is2q)** > > happy new year! here is a fun math fact ---
**Author:** ![Zyada](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zyada/32/3028_2.png) [@Zyada](https://boards.straightdope.com/u/Zyada)\ **Post date:** [January 3, 2025, 4:25am UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443/13 "2025-01-03T04:25:14Z")
And 45 is the sum of the first 9 digits ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [January 3, 2025, 9:51pm UTC](https://boards.straightdope.com/t/2025-sum-of-the-first-nine-cubes/1012443/14 "2025-01-03T21:51:50Z")
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.