Happy 2025!
Do you know that 2025 is a composite, deficient, evil, odd, square, and powerful number? I collect properties of numbers at my Number Gossip website, where you can also find detailed definitions of these terms. Provocatively, 2025 is also an apocalyptic power, meaning that 2 to the power of 2025 contains 666 as a substring.
Recently, Tamas Fleischer sent me an email discussing additional fascinating properties of 2025. While I am slowly deciding whether to add them to my database, there is some urgency in posting these properties in anticipation of the coming year. Here’s the material from Tamas, retold in my own words.
Out of the properties mentioned earlier, the square property is the only rare one. On my website, I define a property as rare if fewer than 100 numbers below 10,000 possess it. Square numbers barely make the cut. But 2025 is not just a square number—it is the square of a triangular number. If you remember the formula for the sum of cubes of the first n natural numbers, the result is (n(n+1)/2)2, which is the square of the nth triangular number. Thus, 2025 is the sum of the cubes of all one-digit numbers.
Additionally, 2025 is the product of 25 and 81. My website notes an intriguing property shared by 25 and 2025: both remain square numbers when all their digits are incremented by 1. For example, 25 becomes 36, and 2025 becomes 3136, both of which are squares. Moreover, 25 is the smallest such number, and 2025 is the second smallest. What my website does not mention is that their square roots exhibit a similar pattern. The square roots of 25 and 2025 are 5 and 45, respectively. When their digits are incremented by 1, the results are 6 and 56, the square roots of 36 and 3136, respectively. The original and incremented squares and their square roots are tied together in a surprising way.
2025 also shares an interesting property with 81. Both are square numbers with an even number of digits and if you split the digits in half and sum the halves, the result is the square root of the original number. For 81, splitting into 8 and 1 gives 8 + 1 = 9, which is the square root of 81. Similarly, for 2025, splitting into 20 and 25 gives 20 + 25 = 45, the square root of 2025. Intriguingly, 81 is the smallest number with this property, and 2025 is the second smallest.
Thank you, Tamas, and Happy New 2025 to everyone!
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