Let $ d (a, b) = 1 – frac {2 gcd (a, b) ^ 3} {ab (a + b)} $ to be a metric on natural numbers.

The metric space $ X = {x_0, x_1, cdots, x_n }, n> 2 $ is isometric embeddable in $ mathbb {R} ^ n $ if and only if the matrix:

$$ M (x_0, x_1, cdots, x_n) = (1/2 (d (x_0, x_i) ^ 2 + d (x_0, x_j) ^ 2-d (x_i, x_j) ^ 2)) _ {1 i, j the n} $$

is semi-definite positive.

So my question is:

Is the above matrix for $ d $ as above semi-defined positive for all

choice of $ x_i in mathbb {N} $? (Perhaps it is possible to prove this using the quadratic method

forms and then transform it into $ sum_ {i} a_ {ii} y_i ^ 2 $ showing then

this $ a_ {ii} $ 0 ge?

If this is the case, it would make Euclidean geometry natural numbers.

For example, for three points / natural numbers (distinct in pairs), we would have:

1) a triangle

2) sine law

3) cosine law

4) All other theorems concerning triangles

Then in the limit three consecutive numbers / prime numbers would construct an equilateral triangle of side length $ 1 $. One could therefore imagine prime numbers ("at the limit") as a simplex of infinite dimension, which would be a fun thing to think about.

Thank you for your help.

Related question:

https://math.stackexchange.com/questions/3385102/is-this-metric-matrix-positive-semidefinite

See Theorem 2.4 in https://books.google.de/books?id=7_DuCAAAQBAJ&printsec=frontcover&hl=de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false for the isometric insertion of $ ( mathbb {N}, d) $ in a Hilbert space.