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: `82361` is 1 less than `82362` (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 `82361` is `6783334321`. Subtracting 1, we get `6783334320`. Note that `6783334320 ÷ 24 = 282638930`.
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: `82361 = 41181^2 - 41180^2 = 1695874761 - 1695792400`
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: `82361 = 100^2 + 269^2 = 10000 + 72361`, where `82361 = 4 × 20590 + 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: `82361` is of the form `4n+1`, where `n = 20590`.
PORTUGUESE
ENGLISH
SPANISH
FRENCH
GERMAN
CHINESE