[seqfan] Relativistic addition of fractions (by young Einstein /; -)

[Tomasz Ordowski]

Dear readers!

Here is the well-known relativistic formula for sum of velocities:
w = (u+v)/(1+uv/c^2). Let the constant c=1 (Planck units).
Note that F(u,v) = (u+v)/(uv+1) = F(1/u,1/v).
Let F(1/n,1/m) = 1/k (unit fractions),
so (nm+1)/(n+m) is an integer.

Let x > 0 be an integer.
I noticed that for a fixed natural n > 1,
the quotient (nx+1)/(n+x) is an integer y
if and only if (n-y)|(ny-1) with 0 < y < n.
The number of solutions y
is d(n^2-1)/2 for n > 1.
Cf. A129296 - OEIS <https://oeis.org/A129296>
(see FORMULA).

Classic (non-relativistic) version.
Let x >= 0 be an integer.
Theorem: for a fixed natural n > 0,
the quotient (nx)/(n+x) is an integer y
if and only if (n-y)|(ny) with 0 <= y < n.
The number of solutions y
is (d(n^2)+1)/2 for n > 0.
See the first comment
on A018892 - OEIS <https://oeis.org/A018892>
(see the NAME).

Best regards,

Thomas Ordowski
_________________________
The relativistic sum of fractions
n/N '+' m/M = (Mn+Nm)/(NM+nm).
In the school formula, nm disappears,
but young Einstein added relativistically...
Alternate arithmetic and new number theory!