# An equivalent form of the Goldbach conjecture using the radical of an integer and the Euler's totient function, and a related problem

Let $\phi(n)$ denoting the Euler's totient function and $\operatorname{rad}(n)$ the so-called radical of an integer (see this Wikipedia.)

The MathWorld's article dedicated to the Goldbach conjecture refers a problem that was proposed working from an equivalent form of the conjecture.

I've written two equivalent forms to the Goldbach conjecture in the same spirit than shows this MathWorld's article, but now using the arithmetic functions $\phi(n)$ and $\operatorname{rad}(n)$. I am going to ask about the statement with a more elaborated (complicated) equation, the other one (more simplest than that I am going to discuss) I presume that is well known or maybe was in the literature (its deduction is simple, the only trick is work with an odd integer in its corresponding $RHS$).

Claim. If $p$ and $q$ are odd primes and $m\geq 2$ is an integer then $$2\operatorname{rad}(p)\operatorname{rad}(q)-\left(\phi(p)\operatorname{rad}(q)+\phi(q)\operatorname{rad}(p)\right)=2m,\tag{1}$$ holds. If additionally for each integer $m\geq 2$ the equation $(1)$ has solution $(p,q,m)$, being $p$ and $q$ odd prime numbers, then the Goldbach conjecture is true.

Thus our motivation was previous MathWorld's article and Claim.

Question. Prove or refute that: For each integer $m\geq 2$ there exist odd integers $x$ and $y$, both greater or equal than $3$, satisfying the (previous) equation $$2\operatorname{rad}(x)\operatorname{rad}(y)-\left(\phi(x)\operatorname{rad}(y)+\phi(y)\operatorname{rad}(x)\right)=2m.\tag{2}$$ Many thanks.

If my Question is very difficult to solve I am considering accept as answer a contribution that provided us remarkable facts about the solution of the equation, well from a theoretical point of view, or well remarkable heuristics or statistics/calculations with a computer about it.

As I've said I didn't add the similar and easy Claim that one can deduce using the same arithmetic functions because I believe that it is known. I believ thus that the equation $(1)$ isn't in the literature. I don't know if my Question is interesting, because I don't know if there are methods to study it, or if my motivation to create this equation $(1)$ and the relationship to the Goldbach conjectue that I've evokes have enough scientific quality.

• Write $(2)$ as $$(\phi(x)-2\,\text{rad}(x))\cdot (\phi(y)-2\,\text{rad}(y)) = 4m+\phi(x)\,\phi(y)$$ and work a bit on this, please. – Jack D'Aurizio Dec 23 '17 at 10:03
• I can from your hints write your equation ( one needs to multiply $(2)$ by two and add $\phi(x)\phi(y)$, then a comparison tell us that I can write my equation as yours). After I believe that the meaning of the factor $\phi(x)-2\operatorname{rad}(x)$ in your equation (and similarly for the other factor in $LHS$) is that: if a prime $p$ in the factorization of $x$ is such that $p^2\mid x$, then such prime $p$ also divides the factor $\phi(x)-2\operatorname{rad}(x)$ (and also divides $\phi(x)\phi(y)$) @JackD'Aurizio – user243301 Dec 27 '17 at 12:57