Happy 2025!

This is a long-overdue New Year's post, and with 2025, I wanted to shed some light on some fascinating mathematical connections with this number. Interestingly, we won't see something like this for precisely another 1000 years (until 3025).

First off, 2025 is a square number. No surprises there, since squares are relatively common when it comes to numbers this size. Still, it's always nice to look at it and know that it has an odd number of factors.

Perhaps more interestingly, it is the sum of the first 9 perfect cubes! In other words, 45^2 = (1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3)! Also, 1 + 2 + 3 + ... + 9 = 45! (not a factorial) That's also quite amazing on its own, and at first glance, it seems completely unrelated to the first fact. But what if I told you that there is another connection between sums of cubes and squares of triangular numbers?

Precisely, that the sum of the first n cubes is exactly the square of the nth triangular number! Amazing, right?

Have a great 2025 and whenever you have a moment, try reflecting on what a rare occurence this is, reminding yourself to use every minute to the fullest.