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: `27529` is 1 more than `27528` (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 `27529` is `757845841`. Subtracting 1, we get `757845840`. Note that `757845840 ÷ 24 = 31576910`.

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: `27529 = 13765^2 - 13764^2 = 189475225 - 189447696`

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: `27529 = 75^2 + 148^2 = 5625 + 21904`, where `27529 = 4 × 6882 + 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: `27529` is of the form `4n+1`, where `n = 6882`.