To let $ text {rad} (x) = prod_ {p | x} p $ be the radical (= product of the prime-number division $ x $) from $ x $ for a natural number $ x $, To the $ x / y> 0 $. $ gcd (x, y) = 1 $ To let $ text {rad} (x / y) = text {rad} (x) cdot text {rad} (y) $, If $ d $ is a metric for nautical numbers, so that $ d (a, b) <1 $ and $ d (a, b) $ is a rational number (I will call such a metric a "rational metric") and for $ epsilon> 0 $ define:

$$ D_ {d, epsilon} (a, b) = 1 – frac {1} { text {rad} ( frac {1} {1 – d (a, b)}) ^ {1+ epsilon}} $$

By an argument similar to the answer of @ GregMartin in the related question, $ D_ {d, epsilon} $ is a metric for natural numbers:

$ D (a, b) = 0 $ iff $ 1 = text {rad} ( frac {1} {1-d (a, b)}) ^ {1+ epsilon} $ iff $ frac {1} {1-d (a, b)} = 1 $ iff $ d (a, b) = 0 $ iff $ a = b $,

The triangular inequality leads to the following inequality:

$$ frac {1} { text {rad} ( frac {1} {1-d (a, c)}) ^ {1+ epsilon}} + frac {1} { text {rad} ( frac {1} {1-d (b, c)}) ^ {1+ epsilon}} = 1+ frac {1} { text {rad} ( frac {1} {1-d ( a, b)}) ^ {1+ epsilon}} $$

This argument comes from @GregMartin:

Both summands are on the left side $ <1/2 $ then the inequality is automatically true because of the $ 1 $ On the right side.

Let the first summand be $ ge 1/2 $, Then

$$ 2 ge text {rad} ( frac {1} {1-d (a, c)}) ^ {1+ epsilon} $$

This can only happen if the right-hand side of the last inequality is $ = 1 $ therefore $ a = c $, But then the inequality to be proved becomes a true equality.

That proves it $ D $ is a metric for natural numbers.

Now consider two rational metrics $ d_1, d_2 $ and let it go $ d (a, b) = d_1 (a, b) + d_2 (a, b) -d_1 (a, b) d_2 (a, b) $

My first question is:

is $ d $ also a rational metric?

If so, you might consider:

The inequality inspired by the related question, and thus by the abc conjecture, is:

$$ max (d_1 (a, b), d_2 (a, b)) <D_ {d, epsilon} (a, b) $$

My second question is:

Can you imagine rational metrics where the above inequality is "easy" to prove?

(Let for example $ d $ be the trivial metric for natural numbers. To let $ d_1 (a, b) = d (a, b) / (d (a, 1) + d (b, 1) + d (a, b)) $ be a $ 1/2 $ Turn stone house from $ d $, which is then a rational metric, and similarly let $ d_2 (a, b) = d (a, b) / (d (a, 2) + d (b, 2) + d (a, b)) $ and define $ d_ {12} = d_1 + d_2-d_1 d_2 $, I wonder if $ d_ {12} $ is a rational metric and if the above inequality is easy to prove or wrong?)

Thanks for your help!

Connected:

A reinterpretation of the $ abc $ conjecture in metric spaces?