PFEL of ratio is difference of PFEL

Dependencies:

1. Prime Factorization Exponent List (PFEL)

PFEL of $\frac{a}{b}$ equals the element-wise difference of PFELs of $a$ and $b$.

Proof

\begin{align} & \operatorname{PFEL}(a) = [a_i]_{i=1}^\infty \wedge \operatorname{PFEL}(b) = [b_i]_{i=1}^\infty \\ &\Rightarrow a = \prod_{i=1}^\infty p_i^{a_i} \wedge b = \prod_{i=1}^\infty p_i^{b_i} \\ &\Rightarrow \frac{a}{b} = \prod_{i=1}^\infty p_i^{a_i - b_i} \\ &\Rightarrow \operatorname{PFEL}\left(\frac{a}{b}\right) = [a_i - b_i]_{i=1}^\infty \end{align}

Dependency for:

1. Divisible iff PFEL is less than or equal

Info:

• Depth: 5
• Number of transitive dependencies: 6

Transitive dependencies:

1. Every number has a prime factorization
2. Integer Division Theorem
3. GCD is the smallest Linear Combination
4. Euclid's lemma
5. Fundamental Theorem of Arithmetic
6. Prime Factorization Exponent List (PFEL)