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: `26209` is 1 more than `26208` (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 `26209` is `686911681`. Subtracting 1, we get `686911680`. Note that `686911680 ÷ 24 = 28621320`.

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: `26209 = 13105^2 - 13104^2 = 171741025 - 171714816`

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: `26209 = 72^2 + 145^2 = 5184 + 21025`, where `26209 = 4 × 6552 + 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: `26209` is of the form `4n+1`, where `n = 6552`.