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: `42709` is 1 more than `42708` (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 `42709` is `1824058681`. Subtracting 1, we get `1824058680`. Note that `1824058680 ÷ 24 = 76002445`.
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: `42709 = 21355^2 - 21354^2 = 456036025 - 455993316`
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: `42709 = 105^2 + 178^2 = 11025 + 31684`, where `42709 = 4 × 10677 + 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: `42709` is of the form `4n+1`, where `n = 10677`.
PORTUGUESE
ENGLISH
SPANISH
FRENCH
GERMAN
CHINESE