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: `29453` is 1 less than `29454` (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 `29453` is `867479209`. Subtracting 1, we get `867479208`. Note that `867479208 ÷ 24 = 36144967`.

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: `29453 = 14727^2 - 14726^2 = 216884529 - 216855076`

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: `29453 = 67^2 + 158^2 = 4489 + 24964`, where `29453 = 4 × 7363 + 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: `29453` is of the form `4n+1`, where `n = 7363`.