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: `39509` is 1 less than `39510` (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 `39509` is `1560961081`. Subtracting 1, we get `1560961080`. Note that `1560961080 ÷ 24 = 65040045`.
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: `39509 = 19755^2 - 19754^2 = 390260025 - 390220516`
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: `39509 = 103^2 + 170^2 = 10609 + 28900`, where `39509 = 4 × 9877 + 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: `39509` is of the form `4n+1`, where `n = 9877`.
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