A generalization of the abc-conjecture?

Question

For a natural number $a$ define $X_a := \{a/k| 1\le k \le a \}$. Then it is not difficult to show that $|X_a \cap X_b| = \gcd(a,b)$. Using this one can define similarities over the natural numbers, using similarities which are defined over finite sets, for example from the Encyclopedia of Distances, replacing $|X_a \cap X_b|$ with $\gcd(a,b)$ and $|X_a|$ with $a$.

Example of such similarities are:

1) $s_{Si}(a,b) = \frac{\gcd(a,b)}{\min(a,b)}$ = Simpson similarity

2) $s_{BB}(a,b) = \frac{\gcd(a,b)}{\max(a,b)}$ = Braun,Blanquet similarity

3) $s_{J}(a,b) = \frac{\gcd(a,b)}{a+b-\gcd(a,b)}$ = Jaccard similarity

4) $s_{S}(a,b) = \frac{2\gcd(a,b)}{a+b}$ = Sorensen similarity

5) $s_{Cos}(a,b) = \frac{\gcd(a,b)^2}{ab}$ = Squared Cosine similarity

They all ($s$ is a similarity from above) have these properties:

i) For all $a,b$ we have $s(a,b) \in \mathbb{Q}$

ii) For all $a,b,m$ we have $s(ma,mb) = s(a,b)$

iii) There exists a natural number $S$ such that for all $a,b$ we have $\frac{S}{s(a,b)} \in \mathbb{N}$

iv) $s(1,1)=1$

The point iii) is not fullfilled for example by the Kulczynski similarity:

$s_K(a,b) = \frac{(a+b)\gcd(a,b)}{2ab}$

To not repeat the definitions I will call similarities on natural numbers with $i)-iv)$ for short $\gcd$-similarities.

Let $s$ be a gcd similarity. Then $s$ is called reducible if there exist gcd-similarities $s_1,s_2$ such that for all $a,b$ we have:

$$\frac{s(a,b)}{S} = \frac{s_1(a,b)}{S_1} \frac{s_2(a,b)}{S_2}$$, where $S,S_1,S_2$ are minimal numbers with property iii).

Otherwise $s$ is callled irreducible.

For instance: $s_{Cos} = s_{Si} \cdot s_{BB}$, hence $s_{Cos}$ is reducible.

I could prove that the Simpson (min), Braun-Blanquet (max), Jaccard and Sorensen similiarties are irreducible.

Why is this interesting anyway?

This irreducible similarties play a role in the formulation of the abc-conjecture, which could be a coincidence of course, which is why I am asking this question, to find out if it is a coincidence or not:

Let $s_1 = s_{S}, s_2 = s_{Si}, s_3 = s_{BB}, S_1 = 2, S_2 = 1, S_3 = 1$.

If $a,b \in \mathbb{N}$, such that we have ("pairwise-gcd-assumption"):

$$ \gcd( \frac{S_i}{s_i(a,b)}, \frac{S_j}{s_j(a,b)}) = 1, \text{ for } i \neq j, 1 \le i,j \le 3$$

Then the abc-conjecture is equivalent to: For all $\epsilon > 0$ there exists $K$ such that for all $a,b$ satisfying the pairwise-gcd-assumption, we have:

$$\max_{i} \{ \frac{S_i}{s_i(a,b)}\} < K \text{rad}( \prod_{1 \le i \le 3} \frac{S_i}{s_i(a,b)} )^{1+\epsilon}$$

Also notice that if $s_1,s_2,s_3$ are such as above, and $d(a,b) = 1-s_1(a,b)s_2(a,b)s_3(a,b)$, then it seems like $d$ can be embedded in Euclidean Space, which is a very strong condition, and very unusual.

My question is, if you can help me test the related conjectures numerically for $\epsilon = 1, K_{\epsilon} = 2$:

Here is some Sage Code which I have used to test it. It would be nice, if you pick a specific $s = s_1 s_2 s_3$ , $s_i \in \{ s_S,s_{Si},s_{BB},s_{J} \}$ and test the corresponding (there are $\binom{4}{3} = 4$ of them) conjecture to find counterexamples.)

Thanks for your help!

def sS(a,b):
    """Sorensen"""
    return 2*gcd(a,b)/(a+b)

def sJ(a,b):
    """Jaccard"""
    return gcd(a,b)/(a+b-gcd(a,b))

def sSi(a,b):
    """Simpson"""
    return gcd(a,b)/min(a,b)

def sBB(a,b):
    """Braun-Blanquet"""
    return gcd(a,b)/max(a,b)


IrredSim = [sS,sJ,sSi,sBB]

d = { sS:2, sJ:1,sSi:1,sBB:1,sL:3}

X = range(1,101)
for s1 in IrredSim:
    for s2 in IrredSim:
        for s3 in IrredSim:
            if len(set([s1.__doc__,s2.__doc__,s3.__doc__]))==3:
                print s1.__doc__,s2.__doc__,s3.__doc__
                for a in X:
                    for b in X:
                      S1 = s1(a,b)/d[s1]
                      S2 = s2(a,b)/d[s1]
                      S3 = s3(a,b)/d[s1]
                      m = max(1/S1,1/S2,1/S3)
                      if m >= 2*rad(1/(S1*S2*S3))^2 and gcd(1/S1,1/S2)==1 and gcd(1/S1,1/S3)==1 and gcd(1/S2,1/S3)==1:
                          print "counterexample:" , s1.__doc__,s2.__doc__,s3.__doc__, a,b

Here are the three combinations, by property iii) we can assume that $\gcd(a,b)=1$:

1) $s_1=s_{Si}, s_2=s_{BB},s_3=s_S$ (abc-conjecture) I think this has been thoroughly tested on the computer ...

2) $s_1=s_{Si}, s_2=s_{BB},s_3=s_J$ If $\gcd(\min(a,b),a+b-1)=1$ and $\gcd(\max(a,b),a+b-1)=1$ then:

$$ a+b-1 < 2 \cdot \text{rad}(ab(a+b-1))^2$$ where we have used $\min(a,b)\max(a,b)=ab$.

3) $s_1=s_{BB}, s_2=s_{S},s_3=s_J$ If $\gcd(\max(a,b),a+b-1)=1$ then:

$$ a+b < 2 \cdot \text{rad}(\max(a,b)(a+b)(a+b-1))^2$$

4) $s_1=s_{Si}, s_2=s_{S},s_3=s_J$ If $\gcd(\min(a,b),a+b-1)=1$ then:

$$ a+b < 2 \cdot \text{rad}(\min(a,b)(a+b)(a+b-1))^2$$

It would be nice if someone checks these conjectures on his/her computer.