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: `26249` is 1 less than `26250` (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 `26249` is `689010001`. Subtracting 1, we get `689010000`. Note that `689010000 ÷ 24 = 28708750`.

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: `26249 = 13125^2 - 13124^2 = 172265625 - 172239376`

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: `26249 = 40^2 + 157^2 = 1600 + 24649`, where `26249 = 4 × 6562 + 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: `26249` is of the form `4n+1`, where `n = 6562`.