User blog:Ynought/Function with unsolved problem

Function with unsolved problem
\(a+b=c\) where \(a,b|\nexists(k|k\in\mathbb{N},\frac{a}{k}\in\mathbb{N}\land\frac{b}{k}\in\mathbb{N})\)

let \(\text{rad}(a,b,c)\) be solved by: \(p=\text{rad}(a,b,c)^k>c\)
 * 1) break a down into its prime factors
 * 2) if a prime appears multiple times then erase just so many of that prime that there is only one left(do as many times as necesary with all primes)
 * 3) multiply those together call it \(\text{rad}(a)\)
 * 4) break b down into its prime factors
 * 5) if a prime appears multiple times then erase just so many of that prime that there is only one left(do as many times as necesary with all primes)
 * 6) multiply those together call it \(\text{rad}(b)\)
 * 7) break c down into its prime factors
 * 8) if a prime appears multiple times then erase just so many of that prime that there is only one left(do as many times as necesary with all primes)
 * 9) multiply those together call it \(\text{rad}(c)\)
 * 10) then let \(\text{rad}(a)\times\text{rad}(b)\times\text{rad}(c)=\text{rad}(a,b,c)\)

the number of exceptions for \(p\) for \(k=n\) and \(a\) and \(b\) can be what they want,as long as they are following the rules above,is called \(p(n)\) (if \(p(n)<\omega\) for \(n>1\) is true is the problem)

\(\begin{eqnarray*} \text{mod}(a,b) = \left\{ \begin{array}{ll} \text{mod}(\text{mod}(a-b),b) & \text{if:}(a-b)\geq 0 \\ a & \text{if:}(a-b)<0 \\ a & \text{if:}b=0 \end{array} \right. \end{eqnarray*}\)

\(\begin{eqnarray*} \text{mod+}(a,b,c) = \left\{ \begin{array}{ll} \text{mod}(\text{mod}(a-b,b,c+a),b,c+a) & \text{if:}(a-b)\geq 0\land a\neq c \\ \mod(a+c,b+a,c+1) & \text{if:}(a-b)<0\land a\neq c \\ a+c & \text{if:}b=0\land a\neq c \land a-c>0 \\ \lfloor \frac{a+c}{2} \rfloor & \text{if:}b=0\land a\neq c \land a-c<0 \\ a+b+c & (a-b)<0\land a=c \\ a+b+c & (a-b)=0\land a=c \end{array}\right. \end{eqnarray*}\)

let \(d(n)=\text{min}(x|\forall_{i,j,k\leq n\land i,j,k\in\mathbb{N}}x>\text{mod+}(i,j,k)\)

\(q(a,0)=d(n)\)

\(q(a,b+1)=min(x|\forall_{kg(d(p(k)),b)\)

Hope you like it :) Feedback is appreciated