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: `63409` is 1 more than `63408` (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 `63409` is `4020701281`. Subtracting 1, we get `4020701280`. Note that `4020701280 ÷ 24 = 167529220`.
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: `63409 = 31705^2 - 31704^2 = 1005207025 - 1005143616`
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: `63409 = 153^2 + 200^2 = 23409 + 40000`, where `63409 = 4 × 15852 + 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: `63409` is of the form `4n+1`, where `n = 15852`.
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