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: `12497` is 1 less than `12498` (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 `12497` is `156175009`. Subtracting 1, we get `156175008`. Note that `156175008 ÷ 24 = 6507292`.

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: `12497 = 6249^2 - 6248^2 = 39050001 - 39037504`

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: `12497 = 41^2 + 104^2 = 1681 + 10816`, where `12497 = 4 × 3124 + 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: `12497` is of the form `4n+1`, where `n = 3124`.