Prime Number Calculator

1) Relation to multiples of 6: Every prime number greater than 3 is always one less or one more than a multiple of 6.
Example: `25657` is 1 more than `25656` (multiple of 6).

2) Square minus 1 divisible by 24: For every prime number greater than 3, if 1 is subtracted from its square, the result is always divisible by 24.
Example: The square of `25657` is `658281649`. Subtracting 1, we get `658281648`. Note that `658281648 ÷ 24 = 27428402`.

3) Difference of squares: Every prime number greater than 2 can be expressed as the difference of the squares of two consecutive natural numbers.
Example: `25657 = 12829^2 - 12828^2 = 164583241 - 164557584`

4) Sum of two squares: Every prime number of the form `4n+1` (where n is a positive natural number) can be expressed as the sum of two squares.
Example: `25657 = 76^2 + 141^2 = 5776 + 19881`, where `25657 = 4 × 6414 + 1`.

5) Form `4n+1` or `4n-1`: Except for the number 2, all prime numbers follow the pattern `4n+1` or `4n-1`, where `n` is a natural number.
Example: `25657` is of the form `4n+1`, where `n = 6414`.