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

[Tomasz Ordowski]

P.S. Two examples.
Classic: 1/3 + 1/6 = 1/2.
Relativistic: 1/3 '+' 1/5 = 1/2.
I recommend this new arithmetic!
Relativistically, only proper fractions can be added.
Then their sum will make physical sense (SR).
However, the mathematical sense is wide.

T. Ordowski

czw., 23 wrz 2021 o 10:39 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> 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!
>
>