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: `12577` is 1 more than `12576` (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 `12577` is `158180929`. Subtracting 1, we get `158180928`. Note that `158180928 ÷ 24 = 6590872`.

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: `12577 = 6289^2 - 6288^2 = 39551521 - 39538944`

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: `12577 = 16^2 + 111^2 = 256 + 12321`, where `12577 = 4 × 3144 + 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: `12577` is of the form `4n+1`, where `n = 3144`.