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: `78649` is 1 more than `78648` (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 `78649` is `6185665201`. Subtracting 1, we get `6185665200`. Note that `6185665200 ÷ 24 = 257736050`.
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: `78649 = 39325^2 - 39324^2 = 1546455625 - 1546376976`
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: `78649 = 140^2 + 243^2 = 19600 + 59049`, where `78649 = 4 × 19662 + 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: `78649` is of the form `4n+1`, where `n = 19662`.
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