20-25 Interesting Facts about 2025

Katie Steckles

Photo by Matheus Bertelli on Pexels

For a mathematician like me, each new year brings a new set of digits to play around with – and it turns out that this year the digits are particularly interesting! The number 2025 has a lot of interesting properties, and here we present somewhere between 20 and 25 (aka 20-25) interesting facts about the number 2025.

  1. It is a square number

One immediately interesting property of 2025 is that it is a square number: 2025 = 45². This is a type of number which gets rarer for larger numbers (there are 31 square numbers below 1000, but between 2000 and 3000 there are only 10, of which 2025 is one). The previous square-numbered year was 44² = 1936, and the next one will be 46² = 2116.

  1. The number it is a square of can be made from the digits of 2025

As an addendum to the previous fact (but definitely a separate fact for accounting purposes), the number 45, which can be squared to get 2025, is itself the result of adding 20 + 25. This is a relatively rare property of pairs of two-digit numbers: the only others which work are 9801 = (98 + 01)², and 3025 = (30 + 25)².

  1. It is the square of a triangular number

Another great fact about 45 is that it is a triangular number: it can be written as the sum of all the numbers from 1 to 9. This means that we can write \(2025 = (1+2+3+4+5+6+7+8+9)^2\).

  1. This means it is also a sum of cubes

Due to a pleasing fact about how squares and cubes are related, any number that can be written as a square of the sum of the first \(n\) numbers (called a triangular number, since that many dots could be drawn in a triangle shape) can also be written as the sum of the cubes of those same numbers. In general, the formula for the sum of the first \(n\) numbers is given by:

\[ 1+2+3+ \cdots + n = \frac{n(n+1)}{2}\]

The sum of the cubes of the first \(n\) numbers is given by:

\[ 1+2+3+ \cdots + n = \frac{n^2(n+1)^2}{4}\]

These formulae were known to mathematicians since antiquity – the Indian mathematician Aryabhata from around 476 CE included both these formulae in his book the Aryabhatiya, and noted that one was the square of the other (see page 37 in this English translation with notes, by Walter Eugene Clark).

The most important consequence of this is that we can also write  \(2025 = 1^3 +2^3 +3^3 +4^3 +5^3 +6^3 +7^3 +8^3 +9^3\).

5, 6 and 7. It really just wants to be square

The number 2025 is itself square, but it remains square if you add together its digits: \(2 + 0 + 2 + 5 = 9 = 3^2\); if you remove its most significant digit, you get 025, which is also a square number \(25 = 5^2\), and if you remove its second digit you get 225, which is \(15^2\).

  1. It is not the sum of two primes

2025 is a square number which cannot be written as a sum of two prime numbers. While this might sound like a non-fact, it is a reasonably rare property of square numbers (only 6 of the 31 squares below 1000 have this property), and it relates to an important open mathematical result: the Goldbach Conjecture. This posits that every even number (greater than 2) can be written as a sum of two prime numbers, and while much progress has been made, the conjecture remains unproven at time of writing. And if it does fail, it will be the fault of numbers like 2025…

  1.  It is one more than a tetrahedral number

Another notable fact about 2025 (which is technically an interesting fact about 2024, but we will allow it) is that it immediately follows a tetrahedral number: sometimes also called a cannonball number, these are numbers which form triangular pyramids, or tetrahedra, where each layer is a triangle larger than the one above it. (If you have that number of cannonballs, they can be stacked neatly in a pyramid). These numbers are formed as sums of consecutive triangular numbers.

A square number (like 2025) which is one more than a tetrahedral number (like 2024) is a very rare occurrence indeed: the previous set of numbers for which it worked was 120 and 121, and as Peter Rowlett notes in his 2025 number facts blog post, that was never actually celebrated as a calendar year, since the current dating system was not invented until 525 CE. So this was the first New year’s Eve it was possible to perform a celebratory tetrahedron-to-square transformation, as attempted by mathematician Max Hughes in their excellent video.

  1. It is the sum of the times tables of the numbers 1 to 9

If you write out all the possible multiplications starting with 1 × 1, 1 × 2, 1 × 3, … ,1 × 9 and so on up to 9 × 9, including each pair of numbers once only, the sum of all of these will be 2025. This can be neatly shown by writing out the multiplications for each factor and rearranging:

\[ (1 \times 1) + (1 \times 2) + \cdots + (1 \times 9) = 1 \times (1+2+3+4+\cdots+9) = 1 \times 45\]

\[ (2 \times 1) + (2 \times 2) + \cdots + (2 \times 9) = 2 \times (1+2+3+4+\cdots+9) = 2 \times 45\]

\[ (3 \times 1) + (3 \times 2) + \cdots + (3 \times 9) = 3 \times (1+2+3+4+\cdots+9) = 3 \times 45\]

\[ \cdots \]

So the overall sum will be

\[  (1+2+3+4+5+6+7+8+9) \times 45 = 45 \times 45 = 2025\]

(Anyone who claims this is the same fact as one of the earlier facts is incorrect – this is definitely a different fact). 

In order to reach the stated goal of 20-25 facts about 20-25, the next several facts will be presented in quick-fire format.

  1. 2025 can be written as the sum of two squares: \( 2025 = 27^2 + 36^2\)
  1. It can be written as the sum of three squares: \( 2025 = 40^2 + 20^2 + 5^2\)
  1. It can be written as the product of two squares: \( 2025 = 5^2 \times 9^2\)
  1. If you multiply together all the numbers which are divisors of 45 (1, 3, 5, 9 and 15) you get 2025
  1. It has lots of connections with the number 15

Suitably for fact number 15, 2025 is the smallest multiple of 15 which has exactly 15 divisors, and all of its divisors are odd (which means it is one of the numbers whose number of odd divisors is also a divisor of the number). It is also divisible by all of the factors of 15.

  1. People born around 1980 will spend at least some of the year \(45^2\) being aged 45.

Much more interesting than celebrating a 40th or 50th birthday!

  1. The year 2025 will also include several mathematically interesting dates

Since 25 is a number found in the Pythagorean triple 7, 24, 25, this year we can celebrate 24/7/25 on 24th July – a truly Pythagorean occasion; and since 25 is \(5^2\) we can also celebrate \(4^2 / 3^2 / 5^2\) on 4th September, representing the squares of the famous 3,4,5 Pythagorean triple.

If you prefer to write the date with a four-digit year but still want to celebrate a Pythagorean coincidence, the date 9/12/2025 (or 12/9/2025) could be an occasion to celebrate a Pythagorean quadruple: since \(9^2 + 12^2 + 20^2 = 25^2\), these four numbers form the sides and diagonal of a 3D cuboid with integers in every direction and along its 3D diagonal.

  1. The sum of any pair of consecutive digits in 2025 is prime

2 + 0 = 2, 0 + 2 = 2, and 2 + 5 = 7. What do you mean, that is unimpressive? This was also true for 2023, 2021 and (actually unimpressively) 2020, but the last year before that for which this was true was 1676.

  1. 2025 is the sum of the odd numbers between 400 and 409 – that is, \(2025 = 401 + 403 + 405 + 407 + 409\)

And, technically satisfying the requirement of ‘20-25 facts’, the final mind-blowing fact about the number 2025 is this:

20. \[ 1^0 + 2^0 + 3^0 + \cdots + 2025^0 = 2025\]

The post 20-25 Interesting Facts about 2025 originally appeared on the HLFF SciLogs blog.

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