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Given a number $x \in \mathbb{N}$ , I want to write down following algorithm in a notation which can be written without the need for providing an example.

Step (1): Find all unique prime factors $p_1, p_2, ..., p_n$ of $x$ ( $p_1 \neq p_2 \neq ... \neq p_n$ , $p_n \in \mathbb{P}$ , $n \in \mathbb{N}$ ):

$x = p_1^{u_1} \cdot p_2^{u_2} \cdot ... \cdot p_n^{u_n}$

Step (2): ???

Example for $x:=10$

$x := 10 = 2^1 \cdot 5^1 \Rightarrow p_1 = 2$ , $p_2 = 5$

$D_1(10) = \{ m | m \equiv 0\ (mod\ 2) \wedge m \equiv 0\ (mod\ 5) , m \in \mathbb{N}\}$

$D_2(10) = \{ m | m \equiv 0\ (mod\ 2) \wedge m \equiv 1\ (mod\ 5) , m \in \mathbb{N}\}$

$D_3(10) = \{ m | m \equiv 1\ (mod\ 2) \wedge m \equiv 0\ (mod\ 5) , m \in \mathbb{N}\}$

$D_4(10) = \{ m | m \equiv 1\ (mod\ 2) \wedge m \equiv 1\ (mod\ 5) , m \in \mathbb{N}\}$

Example for $x:=30$

$x := 30 = 2^1 \cdot 3^1 \cdot 5^1 \Rightarrow p_1 = 2$ , $p_2 = 3$ , $p_3 = 5$

$D_1(30) = \{ m | m \equiv 0\ (mod\ 2) \wedge m \equiv 0\ (mod\ 3) \wedge m \equiv 0\ (mod\ 5) , m \in \mathbb{N}\}$

$D_2(30) = \{ m | m \equiv 0\ (mod\ 2) \wedge m \equiv 0\ (mod\ 3) \wedge m \equiv 1\ (mod\ 5) , m \in \mathbb{N}\}$

$D_3(30) = \{ m | m \equiv 0\ (mod\ 2) \wedge m \equiv 1\ (mod\ 3) \wedge m \equiv 0\ (mod\ 5) , m \in \mathbb{N}\}$

$D_4(30) = \{ m | m \equiv 0\ (mod\ 2) \wedge m \equiv 1\ (mod\ 3) \wedge m \equiv 1\ (mod\ 5) , m \in \mathbb{N}\}$

$D_5(30) = \{ m | m \equiv 1\ (mod\ 2) \wedge m \equiv 0\ (mod\ 3) \wedge m \equiv 0\ (mod\ 5) , m \in \mathbb{N}\}$

$D_6(30) = \{ m | m \equiv 1\ (mod\ 2) \wedge m \equiv 0\ (mod\ 3) \wedge m \equiv 1\ (mod\ 5) , m \in \mathbb{N}\}$

$D_7(30) = \{ m | m \equiv 1\ (mod\ 2) \wedge m \equiv 1\ (mod\ 3) \wedge m \equiv 0\ (mod\ 5) , m \in \mathbb{N}\}$

$D_8(30) = \{ m | m \equiv 1\ (mod\ 2) \wedge m \equiv 1\ (mod\ 3) \wedge m \equiv 1\ (mod\ 5) , m \in \mathbb{N}\}$

For step (2) I have problems describing this binary permutation in natural language, and in a mathematical notation, without providing an example. I am searching for a mathematical optimal notation and easy understandable description. I would be glad if you could help me with this.

Daniel Marschall
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  • Why only binaries? How would you represent $4=2^2$? And where did the $x$ go in your descriptions? – draks ... Jun 17 '15 at 07:50
  • It's not clear from your example what you're trying to describe. Can you explain the desired outcome for $x = 4$ and $x = 30$, for instance? – Daniel McLaury Jun 17 '15 at 08:17
  • In step (1), I find the unique prime factors of $x$ . $x$ is an unknown number $x \in \mathbb{N}$ which has to be processed by the algorithm. In case that $x = 4 = 2^2$ , $p_1 = 2$ , and in case $x = 30 = 2^1 \cdot 3^1 \cdot 5^1$ , so $p_1 = 2 , p_2 = 3 , p_3 = 5$ . In step (2), I only need binaries (0, 1). In case of $x=30$, I would need the permutations 000, 001, 010, 011, 100, 101, 110, 111 with $mod 2$ , $mod 3$ , $mod 5$ to form the definitions of $D_1(30) ... D_8(30)$ . I hope it is more clear now. – Daniel Marschall Jun 17 '15 at 08:31
  • I have edited my OP with the example of $x=30$ – Daniel Marschall Jun 17 '15 at 08:41

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