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: `11093` is 1 less than `11094` (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 `11093` is `123054649`. Subtracting 1, we get `123054648`. Note that `123054648 ÷ 24 = 5127277`.

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: `11093 = 5547^2 - 5546^2 = 30769209 - 30758116`

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: `11093 = 22^2 + 103^2 = 484 + 10609`, where `11093 = 4 × 2773 + 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: `11093` is of the form `4n+1`, where `n = 2773`.