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: `31177` is 1 more than `31176` (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 `31177` is `972005329`. Subtracting 1, we get `972005328`. Note that `972005328 ÷ 24 = 40500222`.

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: `31177 = 15589^2 - 15588^2 = 243016921 - 242985744`

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: `31177 = 44^2 + 171^2 = 1936 + 29241`, where `31177 = 4 × 7794 + 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: `31177` is of the form `4n+1`, where `n = 7794`.