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: `12721` is 1 more than `12720` (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 `12721` is `161823841`. Subtracting 1, we get `161823840`. Note that `161823840 ÷ 24 = 6742660`.

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: `12721 = 6361^2 - 6360^2 = 40462321 - 40449600`

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: `12721 = 20^2 + 111^2 = 400 + 12321`, where `12721 = 4 × 3180 + 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: `12721` is of the form `4n+1`, where `n = 3180`.