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: `11801` is 1 less than `11802` (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 `11801` is `139263601`. Subtracting 1, we get `139263600`. Note that `139263600 ÷ 24 = 5802650`.

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: `11801 = 5901^2 - 5900^2 = 34821801 - 34810000`

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: `11801 = 40^2 + 101^2 = 1600 + 10201`, where `11801 = 4 × 2950 + 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: `11801` is of the form `4n+1`, where `n = 2950`.