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: `84181` is 1 more than `84180` (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 `84181` is `7086440761`. Subtracting 1, we get `7086440760`. Note that `7086440760 ÷ 24 = 295268365`.
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: `84181 = 42091^2 - 42090^2 = 1771652281 - 1771568100`
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: `84181 = 9^2 + 290^2 = 81 + 84100`, where `84181 = 4 × 21045 + 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: `84181` is of the form `4n+1`, where `n = 21045`.
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