Together with my friend mick We came to consider the following ( after thinking about fake function theory ) :

Let a given A be 0 < A < 2.

tommy-mick 3 entires Conjecture :

For any real entire f(x) there exist real entire g(x) and h(x) such that

A f(x) = f(g(x)) + f(h(x))

or

A f(x) = f(g(x)) - f(h(x))

is satisfied.

The idea of positive derivatives and the alike led to many disputes

( and related to fake function theory naturally )

Im tempted to mention that I considered some matrix ideas such as Carleman and some ideas from Gottfried.

( Though the ideas I tried based on Gottfried's ideas seem to have trouble with convergeance ... )

But with no result so far.

Its possible that I posted this before in the past and maybe it has been considered 100 years ago.

I believe the real intresting things happen when we develop ( or rediscover ) the tools to solve this.

This is because this problem is a cornerstone ; many numerical methods and theoretical ideas I had - but did not publish - depend on the truth of this and a way to compute it.

Bye the way you can vote for mick on MSE , he put himself as a candidate for moderator.

Since he is probably the only young MSE user into tetration with a rep above 2000 he might be the man ( boy ) to defend/judge tetration Q&A.

My apologies if im wrong : maybe some other users here or on MSE are young and have rep above 2000 too.

( Sheldon and Gottfried are on MSE but they are older , perhaps im forgetting balarka or galathaea or ... )

regards

tommy1729

Let a given A be 0 < A < 2.

tommy-mick 3 entires Conjecture :

For any real entire f(x) there exist real entire g(x) and h(x) such that

A f(x) = f(g(x)) + f(h(x))

or

A f(x) = f(g(x)) - f(h(x))

is satisfied.

The idea of positive derivatives and the alike led to many disputes

( and related to fake function theory naturally )

Im tempted to mention that I considered some matrix ideas such as Carleman and some ideas from Gottfried.

( Though the ideas I tried based on Gottfried's ideas seem to have trouble with convergeance ... )

But with no result so far.

Its possible that I posted this before in the past and maybe it has been considered 100 years ago.

I believe the real intresting things happen when we develop ( or rediscover ) the tools to solve this.

This is because this problem is a cornerstone ; many numerical methods and theoretical ideas I had - but did not publish - depend on the truth of this and a way to compute it.

Bye the way you can vote for mick on MSE , he put himself as a candidate for moderator.

Since he is probably the only young MSE user into tetration with a rep above 2000 he might be the man ( boy ) to defend/judge tetration Q&A.

My apologies if im wrong : maybe some other users here or on MSE are young and have rep above 2000 too.

( Sheldon and Gottfried are on MSE but they are older , perhaps im forgetting balarka or galathaea or ... )

regards

tommy1729