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: `13693` is 1 more than `13692` (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 `13693` is `187498249`. Subtracting 1, we get `187498248`. Note that `187498248 ÷ 24 = 7812427`.

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: `13693 = 6847^2 - 6846^2 = 46881409 - 46867716`

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: `13693 = 2^2 + 117^2 = 4 + 13689`, where `13693 = 4 × 3423 + 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: `13693` is of the form `4n+1`, where `n = 3423`.