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: `38821` is 1 more than `38820` (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 `38821` is `1507070041`. Subtracting 1, we get `1507070040`. Note that `1507070040 ÷ 24 = 62794585`.
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: `38821 = 19411^2 - 19410^2 = 376786921 - 376748100`
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: `38821 = 65^2 + 186^2 = 4225 + 34596`, where `38821 = 4 × 9705 + 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: `38821` is of the form `4n+1`, where `n = 9705`.
PORTUGUESE
ENGLISH
SPANISH
FRENCH
GERMAN
CHINESE