The math of 2025 

 

Written by: Hugo Medina, Yani Fodil, David Vadillo and Bruno Esteban. 

 

We are in 2025, and perhaps it does not look like a special year, mathematically 2025 is a very wonderful number, starting with the fact that the calendar repeats itself every 50 years so, this year is the same as 1975. 

2025 is made by a perfect square, what is a number made from the multiplication of other number two times itself, and 2025 is equal to 45², that means that 2025 is equal to 45 times 45, that makes it a special year because the last time it happened it was in 1936, that is equal to 44², and it won’t happen until 2116, equal to 46², the same way that 45² is equal to 2025 and 45 is equal to the sum of all the numbers from the 0-9, 0+1+2+3+4+5+6+7+8+9, so 2025 is equal to the square of the summary of the first 10 numbers, that is (1+2+3+4+5+6+7+8+9)², another way to write 45². Another way to see it, that is very single, that way is made by the square of 20 plus 25, (20+25)². 

 

2025 is also the summary of the cube of the same numbers, that means that   0³+1³+2³+3³+4³+5³+6³+7³+8³+9³ is also equal to 2025, but that’s not all, there are some numbers that can be made by x² times y², that is the product of a perfect squares, and 2025 is the product of 9² * 5², that casually it can be written like the square of the product of all the numbers plus the square of the subtraction of the two first number minus the other two ones, (2+0+2+5)² * (20-25)² , also 2025 can be written like the sum of three perfect squares,  5² + 20² + 40², also with the sum of two perfect squares, 27²+36², that at the same time are also the sum of two number elevated to four 9⁴ + 6⁴, that is equal to 3 times the sum of 3 elevated to 4 and 2 elevated to 4, 3(2⁴ + 3⁴) 

 

If we add 1111 to 2025 you get 3136, and if you add 11 to 45 you get 56, and casually the same way that 2025 is equal to 45², 3136 is equal to 56², so 2025 + 1111 = (45 + 11)², that is the same to 3136 = 56², another curious fact, is that the sum of the divisors of 2025 is equal to a perfect square; 76², the way that 2025 + 675 + 405 + 225 + 135 + 81 + 75 + 45 + 27 + 25 + 15 + 9 + 5 + 3 + 1 = 5776 = 76². 

 

Now, we have the theorems and people’s numbers, starting with another interesting feature, that 2025 is a Hamming number or 5-soft number, because all its prime factors (3 and 5) are less than or equal to 5. These types of numbers have applications in computer science or even in musical tuning, particularly in the right temperament, which seeks to maintain precise harmony. 

 

2025 is also in the Zeckendorf theorem, which means that is the result of the sum of not consecutive Fibonacci numbers, one example is with the Fibonacci numbers F17+F14+F9+F7+F4+F2, that are F17 = 1597 + F14 = 377 + F9 = 34 + F7→13 + F4 = 3 + F2 = 1, so substituting the number we get this: 1597 + 377 + 34 + 13 + 3 + 1 =2025. 

 

2025 is a centered octagonal number is a centered figurative number that represents an octagon with a point at the center and all other points surrounding the center point in successive octagonal layers, it is describe such (2n-1) ² or 4n²-4n+1, and if you substitute n = 23 (the 23rd octagon) the number of pints is 2025. 

 

2025 is also a Harshad-Niven number and a Harshad-Niven number is a number that is divisible by the sum of its digits, so we have that 2 + 0 + 2 + 5 = 9 and 2025 is divisible to 9, and is equal to 225, 2025/9=225. 

 

Now, related to the Pythagoras theorem, 2025² can be written such the sum of 567²+1944² or 1215²+1620², both are equal to 2025², so 567²+1944² = 2025² = 1215²+1620².