03/27/2015, 11:20 PM

What I meant is that you have tetration !

Not just for the bases between 1 and eta but for all real bases larger than 1.

Not sure if you realise it yet.

Here is a sketchy way to show it :

In short since you can interpolate analyticly x^x^... = m

where ... are integer iterations and m is a real > e ...

You can THUS solve for the x in tet_x(t) = m for a given m > e and t >0.

( x is the base ).

But this also means that you can solve for t since you can set up the equation RAM(m,t) = x for any desired x.

WHen you have this t , you have found tet_base_x(t) = m for a given m.

In other words from the relation tet_x(t) = m you can solve for either x or t.

therefore you can solve sexp_x(t) = m

which is slog_x(m).

Then invert this function and you have sexp for any base > 1.

Since all of this is done analyticly you have found tetration.

And it seems simpler then some other methods , like Kneser or Cauchy.

Hope this is clear enough.

I can explain more if required.

SO JmsNxn finally has his own method , with credit to the brilliant comment of fivexthethird.

( Im thinking of a variant of this method too )

I just wonder what this will be called ...

JMS method ? JN method ? Jms5x3 method ?

Jms5x31729 method

I already started calling it in my head " Ramanujan-Lagrange method ".

The reason seems clear : Ramanujan's master theorem and Lagrange's inversion theorem.

For those unfamiliar :

http://en.wikipedia.org/wiki/Lagrange_inversion_theorem

regards

tommy1729

Not just for the bases between 1 and eta but for all real bases larger than 1.

Not sure if you realise it yet.

Here is a sketchy way to show it :

In short since you can interpolate analyticly x^x^... = m

where ... are integer iterations and m is a real > e ...

You can THUS solve for the x in tet_x(t) = m for a given m > e and t >0.

( x is the base ).

But this also means that you can solve for t since you can set up the equation RAM(m,t) = x for any desired x.

WHen you have this t , you have found tet_base_x(t) = m for a given m.

In other words from the relation tet_x(t) = m you can solve for either x or t.

therefore you can solve sexp_x(t) = m

which is slog_x(m).

Then invert this function and you have sexp for any base > 1.

Since all of this is done analyticly you have found tetration.

And it seems simpler then some other methods , like Kneser or Cauchy.

Hope this is clear enough.

I can explain more if required.

SO JmsNxn finally has his own method , with credit to the brilliant comment of fivexthethird.

( Im thinking of a variant of this method too )

I just wonder what this will be called ...

JMS method ? JN method ? Jms5x3 method ?

Jms5x31729 method

I already started calling it in my head " Ramanujan-Lagrange method ".

The reason seems clear : Ramanujan's master theorem and Lagrange's inversion theorem.

For those unfamiliar :

http://en.wikipedia.org/wiki/Lagrange_inversion_theorem

regards

tommy1729