A remarkable year

The year 2025 is remarkable not just because some remarkable events are sure to occur but for another reason: 

2025 = 45². 

To be a square number is sufficiently rare that few of us will experience another square in their lifetimes. The last time it happened was in 1936 and the next time will be in 2116. But 2025 has some other numerical features that make it stand out even more. 

 2025 = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³. 

And

2025 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)².

So, remarkably,

1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)²

In general, "9" here can be replaced by any integer. This general result is called Nicomachus's Theorem although I am not sure he formulated it quite like that. There are many proofs and I learnt the easy inductive one at high school in 1961. Some of the geometric proofs are very pretty. There are also very many generalisations.

However, given that the last occurrence of the formula occurred in 1296 and the next will be in 3025, we should celebrate the year 2025 and hope that the world will survive another 1000 years.