This calculator performs the decomposition of a number into its prime factors and lists all its natural divisors. Enter an integer between 1 and 1,000,000.
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: `65393` is 1 less than `65394` (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 `65393` is `4276244449`. Subtracting 1, we get `4276244448`. Note that `4276244448 ÷ 24 = 178176852`.
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: `65393 = 32697^2 - 32696^2 = 1069093809 - 1069028416`
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: `65393 = 143^2 + 212^2 = 20449 + 44944`, where `65393 = 4 × 16348 + 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: `65393` is of the form `4n+1`, where `n = 16348`.
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