There seem to be a lot of numerical coincidences bouncing around concerning the new year 2025. For example, it’s a square number: \( 2025 = 45^2 \). The last square year was \(44^2 = 1936\), and the next will be \(46^2=2116\).

The other one you have likely seen somewhere is this little gem: that 2025 equals both \((1+2+3+4+5+6+7+8+9)^2\) and \(1^3+2^3+3^3+4^3+5^3+6^3+7^3+8^3+9^3\).

But there are more – after all, 2025 does appear in over a thousand search results at the OEIS. Here’s a little collection:

• 2025 is square, composite, deficient, evil, odd and powerful;
• 2025 is an odd refactorable number;
• 2025 is a Lucas 9-step numbers;
• The digit sum of 2025 is also a square, i.e. \(2+0+2+5=9\);
• Any two consecutive digits of 2025 are prime, i.e. \(2+0=2\), \(0+2=2\) and \(2+5=7\);
• The square root of 2025 is 45. If we increment the digits of 45 and square this, something interesting happens: \((45+10+1)^2 = 45^2+1000+100+10+1\), that is if we increment all the digits of 2025 by 1 we obtain a new square number \(3136=56^2\);
• 2025 is the square of a triangular number;
• 2025 remains square when its most significant digit is removed, i.e. 025 is square;
• 2025 is a square which is not the sum of two primes, which is rarer than you may think;
• 2025 is the sum of two squares: \(27^2+36^2\);
• 2025 is the sum of three squares: \(40^2+20^2+5^2\);
• 2025 is the product of two squares: \(9^2 \times 5^2\), it’s also \(3^4 \times 5^2\), a number of the form \(3^i \times 5^j\) with \(i,j>0\);
• \(2025 = (20+25)^2\);
• 2025 is the smallest multiple of 15 with 15 divisors – it’s \(15 \times 135\);
• 2025 is divisible by all the factors of 15;
• 2025 is the smallest number with exactly 15 odd divisors, it’s also notable that 2025 is a number where its number of odd divisors is itself an odd divisor;
• 2025 is the product of the proper divisors of its square root, 45, i.e. \(1 \times 3 \times 5 \times 9 \times 15\);
• 2025 can be written in the form \((4n+1)^2\) for \(n=11\) (there are lots similar to this on OEIS!);
• If you split 2025 in 20 and 25, first these are consecutive digits in the five times table, but also their sum is 45, which is the square root of 2025 (it’s one of only three four-digit numbers with this property);
• \(2025 = (5 \times 9)^2\);
• \(\frac{2025}{9}\) is a square – \(225 = 15^2\);
• The floor of \(\frac{2025}{7}\) is a square – \(289 = 17^2\);
• 2025 is the fifth smallest square that begins ‘2’, the smallest square that begins ’20’ and the smallest square that begins ‘202’;
• 2025 is the sum of the odd numbers from 400 to 409, i.e. \(401+403+405+407+409\);
• There are 2025 strict integer partitions of 45 with no part divisible by all the others;
• There are 2025 partitions of 45 into 3 or more distinct parts;
• 2025 is the product \(\prod_{k=0}^n k^2+(n-k)^2\) for \(n=3\), meaning it can be written \((0^2 + 3^2) \times (1^2 + 2^2) \times (2^2 + 1^2) \times (3^2 + 0^2)\);
• There are nontrivial positive magic squares of six different orders with magic sum 2025;
• 2024 and 2025 is the first time we’ve had a square year following a tetrahedral year (the last time this happened was 121 following 120, but the BC/AD dating system wasn’t invented until 525) – the next time this happens will be 2600 and 2601 (thanks to Max Hughes in the TMiP chat – Max is planning to post a related video on their @max-matics YouTube channel);
• People born in 1980 will be 45 years old in the year \(45^2\);
• If you write out the multiplication table from \(1 \times 1\) to \(9 \times 9\) and sum the entries, you get 2025;
• $645 \pi \approx 2025$ (thanks to Bruce in the Finite Group chat).

Other posts used as sources but also worth checking out: Happy 2025! at Tanya Khovanova’s Math Blog; A Banner Year at Futility Closet. And check out Tom Edgar’s video with two visual proofs of Nicomachus’s Theorem using 2025 as the demo.