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: `11969` is 1 less than `11970` (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 `11969` is `143256961`. Subtracting 1, we get `143256960`. Note that `143256960 ÷ 24 = 5969040`.

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: `11969 = 5985^2 - 5984^2 = 35820225 - 35808256`

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: `11969 = 65^2 + 88^2 = 4225 + 7744`, where `11969 = 4 × 2992 + 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: `11969` is of the form `4n+1`, where `n = 2992`.