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: `12689` is 1 less than `12690` (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 `12689` is `161010721`. Subtracting 1, we get `161010720`. Note that `161010720 ÷ 24 = 6708780`.

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: `12689 = 6345^2 - 6344^2 = 40259025 - 40246336`

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: `12689 = 65^2 + 92^2 = 4225 + 8464`, where `12689 = 4 × 3172 + 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: `12689` is of the form `4n+1`, where `n = 3172`.