Pattern Play Math

01/07/2025

2025 Proves the Theorem of Ancient Greek Philosopher Nicomachus
By Christopher Gomez January 4, 2025

When broken down mathematically, 2025 proves Ancient Greek Philosopher Nicomachus’ Theorem of a perfect square. Credits: Public Domain, via Wikimedia Commons.
When mathematically broken down, 2025 proves Ancient Greek Philosopher Nicomachus’ Theorem of a perfect square.
According to mathematician Jérôme Cottanceau, Léonard-de-Vinci High School in Montaigu, France, the year 2025 is the square of 45, the sum of all odd numbers up to 89, the sum of numbers one through nine cubed, and the square of one through nine added. The fact that 2025 is the solution to all of these formulas means it completely fulfills the Nicomachus theory.
“We take a number, multiply it by itself, and we get its square. And if it’s called a square, it’s because we can represent the number in the form of a square. And if we calculate the square of 45, we find 45²=2025,” Cottanceau writes in his blog. He continues, “Thus, the partial sum of all odd numbers from 1 to 89 will be equal to 45², that is, 2025.”
For context, the last time a year was equal to a perfect square was in 1936, which is the square of 44. We will not see another year like this until 2116. What makes it even rarer, is that the last time a year completely fulfilled the Nicomachus theorem was in 1296, which means we won’t see another year like this until another millennium passes in 3025.
The Nicomachus Theorem and 2025
First discovered by the ancient Greek philosopher Nicomachus of Gerasa, the theory states that the sum of the first odd number and subsequent ones always equals a perfect square. The theory indicates that adding consecutive numbers and squaring the result will yield a perfect square. Additionally, the sum of cubing all odd numbers in a sequence beginning with one equals this same perfect square.
It is a relatively simple theory, but one that demonstrates a unique, fascinating, and elegant relationship between numbers. 2025 is one of the few numbers that interacts with all the formulas contained within the ancient Greek philosopher’s theorem.
Other mathematical peculiarities about 2025
On top of being the first year to fulfill the Nicomachean Theorem in a millennium, it also fulfills some other mathematical rarities. 2025 is an odd refactorable number, which means it is divisible in all of its divisors.
“A number is said to be refactorable when it is divisible by its number of divisors. For example, 18 is a refactorable number, since it has six divisors (1, 2, 3, 6, 9, 18),” said Cottanceau. “The number 2025, with its nine divisors (1, 5, 9, 25, 45, 81, 225, 405, 2025) is also refactorable.”
Also, 45 squared is not only a perfect square but is also a Kaprekar number. A Kaprekar number can be split in two and added up to its square. For example, in the case of 2025, 20+25=45, making it fulfill this nifty mathematical concept.