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: `31477` is 1 more than `31476` (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 `31477` is `990801529`. Subtracting 1, we get `990801528`. Note that `990801528 ÷ 24 = 41283397`.

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: `31477 = 15739^2 - 15738^2 = 247716121 - 247684644`

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: `31477 = 54^2 + 169^2 = 2916 + 28561`, where `31477 = 4 × 7869 + 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: `31477` is of the form `4n+1`, where `n = 7869`.