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: `12049` is 1 more than `12048` (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 `12049` is `145178401`. Subtracting 1, we get `145178400`. Note that `145178400 ÷ 24 = 6049100`.

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: `12049 = 6025^2 - 6024^2 = 36300625 - 36288576`

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: `12049 = 32^2 + 105^2 = 1024 + 11025`, where `12049 = 4 × 3012 + 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: `12049` is of the form `4n+1`, where `n = 3012`.