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: `16477` is 1 more than `16476` (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 `16477` is `271491529`. Subtracting 1, we get `271491528`. Note that `271491528 ÷ 24 = 11312147`.

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: `16477 = 8239^2 - 8238^2 = 67881121 - 67864644`

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: `16477 = 59^2 + 114^2 = 3481 + 12996`, where `16477 = 4 × 4119 + 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: `16477` is of the form `4n+1`, where `n = 4119`.