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

[Tomasz Ordowski]

P.S. The relationship with the hyperbolic function tanh(x)
and its inverse artanh(x) should be noted at the end.
The method used in SR. Physicists know this well.

The relativistic formula w = (u+v)/(1+uv) = tanh(artanh(u)+artanh(v)) with
c = 1.
The equation artanh(1/x) + artanh(1/y) = artanh(1/z) in natural numbers n >
1.
 is equivalent to the equation (x-1)(y-1)(z+1) = (x+1)(y+1)(z-1). Nice!

I waited for someone to notice this on the SeqFan forum.

T. Ordowski

sob., 25 wrz 2021 o 09:24 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

>
> SUPPLEMENT. Two double identities.
> Classic: 1/n = 1/(2n) + 1/(2n) = 1/(n+1) + 1/(n(n+1)) for n > 0.
> Relativistic: 1/n = 1/(2n-1) '+' 1/(2n+1) = 1/(n+1) '+' 1/(n(n+1)-1) for n
> > 1.
>
> T. Ordowski
>
> pt., 24 wrz 2021 o 13:15 Tomasz Ordowski <tomaszordowski at gmail.com>
> napisał(a):
>
>> 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!
>>>
>>>