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: `13649` is 1 less than `13650` (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 `13649` is `186295201`. Subtracting 1, we get `186295200`. Note that `186295200 ÷ 24 = 7762300`.

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: `13649 = 6825^2 - 6824^2 = 46580625 - 46566976`

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: `13649 = 68^2 + 95^2 = 4624 + 9025`, where `13649 = 4 × 3412 + 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: `13649` is of the form `4n+1`, where `n = 3412`.