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: `100109` is 1 less than `100110` (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 `100109` is `10021811881`. Subtracting 1, we get `10021811880`. Note that `10021811880 ÷ 24 = 417575495`.
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: `100109 = 50055^2 - 50054^2 = 2505503025 - 2505402916`
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: `100109 = 190^2 + 253^2 = 36100 + 64009`, where `100109 = 4 × 25027 + 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: `100109` is of the form `4n+1`, where `n = 25027`.
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