# Two questions around the $abc$-conjecture

Let $$d(a,b) = 1-\frac{2 \gcd(a,b)}{a+b}$$, $$d_{ABC}(a,b) = 1-\frac{2\gcd(a,b)^3}{ab(a+b)}$$ be two metrics on natural numbers.

The abc-conjecture can be formulated using these two metrics as:

For every $$\epsilon >0$$ there exists a $$K_{\epsilon}$$ such that:

$$d(a,b) < D_{\epsilon}(a,b):=1-\frac{2}{K_{\epsilon}\text{rad}(\frac{2}{1-d_{ABC}(a,b)})^{1+\epsilon}}$$

Trying make $$D$$ to a metric on natural numbers, leads to $$K_{\epsilon} = \frac{1}{2^\epsilon}$$. One can prove that with this choice of $$K_{\epsilon}$$, $$D$$ is a metric on natural numbers.

Now this is a very explicit version of the abc-conjecture, and for example for $$\epsilon = 1/2$$ this version is wrong but my conjecture is that for $$\epsilon \ge 1$$ this is true.

My question is:

1) There are various results which build upon the abc-conjecture. Is there $$0 < \epsilon < 1$$ needed, or is it enough to have $$\epsilon \ge 1$$?

I have tested this with the Schoenberg-Criterion for some $$\epsilon$$ and some numbers $$n$$:

2) Is the metric $$D_{\epsilon}$$ for every $$\epsilon>0$$ embeddable in Euclidean Space? (Maybe this is too much to ask.)

(This seems very counterintuitive, because of the radical function one would not expect the $$D$$ to be so nice to behave as to embedd in Euclidean Space.)

Edit: Making the following Ansatz:

Suppose that $$\phi(a)$$ is the vector which embedds the natural number $$a$$. Then $$D_1(a,b)^2 = |\phi(a)|^2+|\phi(b)|^2 - 2 \left< \phi(a), \phi(b) \right>$$

If we assume that:

$$\left< \phi(a), \phi(b) \right> = \frac{-2}{R(a,b)^2}(\frac{4}{R(a,b)^2}-2)$$

where $$R(a,b) = \text{ rad }(\frac{ab(a+b)}{\gcd(a,b)^3})$$

then we get, substituting $$a=b$$ that $$|\phi(a)| = \frac{1}{\sqrt{2}}$$

So this is of course no proof, but if $$\phi$$ existed, then the matrix

$$G_n = (\frac{-2}{R(a,b)^2}(\frac{4}{R(a,b)^2}-2))_{1 \le a,b \le n}$$

should be positive definite, which I have checked for $$n$$ up to $$20$$ in Sagemath, and it is so.

So to be more precise about the question 2):

3) Is the matrix $$G_n$$ as defined above, positive definite?

Related question:

The abc-conjecture as an inequality for inner-products?