[LINK] One for the Numerologists and Mystics...

[Antony Barry]

On Tue, Dec 24, 2024 at 9:05 PM David <dlochrin at aussiebb.com.au> wrote:

>      *  2025, itself is a square, i.e. 45 x 45 = 2025
>      *  it is also a product of two-squares, i.e. 9² x 5² = 2025
>      *  it is also the sum of 3-squares, i.e. 40²+ 20²+5²= 2025
>      *  and remarkably it is also the sum of the cubes, of all the single
> digits, from 1 to 9, i.e.1³+2³+3³+4³+5³+ 6³+7³+8³+9³= 2025.


I asked Perplexity AI is there was anything interesting about 2025 with the
following result -

2025 indeed possesses several interesting properties that make it
noteworthy in number theory:

## Mathematical Properties

**Perfect Square**
2025 is a perfect square, specifically 45^2[1][4]. This property alone
makes it significant in number theory, as square numbers often exhibit
unique characteristics.

**Sum of Consecutive Odd Numbers**
2025 can be expressed as the sum of five consecutive single-digit odd
natural numbers: 1 + 3 + 5 + 7 + 9[3]. This property is particularly
interesting as it connects 2025 to other mathematical sequences.[Note by
Tony: Perplexity seems to have got confused!]

**Unique Squaring Property**
2025 is probably the only positive square which remains a square when
adding 1 to each digit (resulting in 3136) as well as when increasing just
the first digit (resulting in 3025)[2]. This unusual characteristic sets
2025 apart from other numbers.

## Number Theoretical Aspects

**Prime Factorization**
The prime factorization of 2025 is 3^4 * 5^2[1][6]. This decomposition into
prime factors is fundamental in number theory and contributes to many of
its other properties.

**Divisibility**
2025 has 15 divisors[1], making it a highly composite number. It is also a
tau number, as it is divisible by the number of its divisors[2].

**Special Number Classifications**
2025 falls into several interesting number categories:
- It is a Harshad number (divisible by the sum of its digits)[2].
- It is a Curzon number[2].
- It is an apocalyptic number[2].
- It is a gapful number (divisible by the number formed by its first and
last digit)[2].

## Unique Representations

**Sum of Squares**
2025 can be written as a sum of two non-zero squares in only one way: 27^2
+ 36^2[2]. This property relates to the Pythagorean theorem and Fermat's
theorem on sums of two squares.

**Sum of Consecutive Primes**
Interestingly, 2025 can be represented as the sum of five consecutive prime
numbers: 397 + 401 + 409 + 419 + 421 = 2025[7]. This connection to prime
numbers adds to its number-theoretical significance.

## Conclusion

These properties collectively make 2025 a number of considerable interest
in number theory. Its status as a perfect square, its unique
representations as sums of consecutive numbers (both odd and prime), and
its various special number classifications demonstrate that 2025 possesses
a rich set of mathematical characteristics worthy of study in number theory.

Sources
[1] Properties of the number 2025 - Number Empire
https://www.numberempire.com/2025
[2] Properties of number 2025 - Numbers Aplenty
https://www.numbersaplenty.com/2025
[3] 25 (number) - Wikipedia https://en.wikipedia.org/wiki/25_(number)
[4] Number 2025 - Math Tools https://math.tools/numbers/2025
[5] Properties of natural numbers such that there is a "very large largest
...
https://mathoverflow.net/questions/101463/properties-of-natural-numbers-such-that-there-is-a-very-large-largest-number-w
[6] Prime factors of 2025 - Math Tools
https://math.tools/numbers/prime-factors/2025
[7] Do you know this The number 2025 can be represented ... - Threads
https://www.threads.net/@data.science.beginners/post/DD7WsWTTuaP
[8] what is special about the number 2025? - hellosunday.ca
https://hellosunday.ca/Vierte4/what-is-special-about-the-number-2025

Tony

>
>
-- 
Mob:04 3365 2400 Email: antonybbarry at gmail.com, antonybbarry at me.com