Re: e^(e^(e^e)) =N ?

On Jan 25, 2:22 pm, VK <schools_r...@xxxxxxxxx> wrote:
PROBLEM FOR COMPUTER SCIENTIST:

Decide if e^(e^(e^e))  is a natural number or not.

e^(e^(e^e))  ~= 2.33150438250040388E1656520

It is not relevant to this NG.

If you're talking about sci.math (in which I am viewing your reply,
and I don't see any other NGs listed in the header), then I beg to
disagree.

Well, if we consider math problems such as
 4.9 * 50 == 245.00000000000003
 (true as true can be)
or
 e == 2.718281828459045(0)
 true as well
or
 e^e = 15.154262241479259(0)
 true as well; due to slightly different IEEE-754 FP-DP
implementations
 the last 2-3 digits will vary by OS and even the same OS versions

- if we consider this as a SCI.math problems then I would agree with
you. If we stay on that within the scope of this group e is an
irrational number, 4.9 * 50 == 245.(0) etc. then the questions like
that one are clearly for COMP.math and not for SCI.math domains.
I am humbly staying on the latter opinion ;-)

Fair enough, the header of the original post did say "PROBLEM FOR
COMPUTER SCIENTIST" I guess. But from a "pure" math perspective, we
want to prove the claim without doing the computation, preferably
using a method that's extensible to similar questions (for example,
prove that exp^n(x) is not rational for any rational x and positive
integer n, where "exp^n" denotes the n-fold iteration of exp).


The discrete computer math operates "in reality" only with dyadic
rationals of a relatively limited min-max range where the range
depends on the word size of the used hardware (8, 16, 32, 64-bit),
stored in IEEE-754 FP-DP format

<snip>

This is all fair enough, but you seem to be overlooking the fact that,
with the right software, the answer can be calculated numerically to
any desired precision, limited only by processor speed and time
available.

True (up to some extends). But the OP was about e - not about 2.718(0)
or 2.71828182(0) or even 2.718281828459045(0)

As the exact value of e cannot be stored by definition on any
computer, no math relevant results - in the sense of sci.math, I am
not talking about practical purposes! - can be ever obtained on any
computer because by the same definition it will deal not with e but
with some finite surrogate values of any desired length.

I don't agree. All that's needed to answer the OP's question
numerically is to evaluate e^(e^(e^e)) to more than 1656520
significant digits. This is a feasible task on a fast enough computer.
In another part of the thread, a poster reports getting an answer in
less than 14 minutes.
.

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