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From: Transfer Principle on 14 Mar 2010 18:34 On Mar 14, 5:54 am, Tony <tempt...(a)freemail.hu> wrote: > On Mar 14, 1:20 am, bill <b92...(a)yahoo.com> wrote: > > On Mar 13, 1:59 pm, Pafnuty Tschebyscheff <th...(a)SDF-EU.ORG> wrote: > > > Is e^e rational? > > > Can anyone give me any references to this problem? > > > Thanks in advance. > > e^e = 1 + e + e^2/2! + e^3/3! ad infinitum. > > Does this help? > No. Today's Pi Day. So what's this thread about that _other_ transcendental number doing here? Even though numbers like e^e and pi^pi are most likely transcendental, I'd personally find it interesting if either were rational (or even algebraic). That's because I'm interested in the operation of tetration (since e^e and pi^pi are just e^^2 and pi^^2). If e^e (respectively pi^pi) turned out to be rational, it would mean that e (respectively pi) would be the super square root (i.e., root of tetration) of a rational number. And of course, if e^e and pi^pi are transcendental, then we can keep trying with e^e^e and pi^pi^pi (i.e., e^^3 and pi^^3) for super cube roots, and so on.
From: Phil Carmody on 24 Mar 2010 19:01 Bart Goddard <goddardbe(a)netscape.net> writes: > Tony <temptony(a)freemail.hu> wrote: >> On Mar 14, 1:20 am, bill <b92...(a)yahoo.com> wrote: >>> On Mar 13, 1:59 pm, Pafnuty Tschebyscheff <th...(a)SDF-EU.ORG> wrote: >>> >>> > Is e^e rational? >>> > Can anyone give me any references to this problem? >>> > Thanks in advance. >>> >>> e^e = 1 + e + e^2/2! + e^3/3! ad infinitum. >>> >>> Does this help? >> >> No. > > An old "West Coast Number Theory" problem, (before 1980, > I think) asks for the existence of a "Humdrum Number" > which was defined as an integer N such that e^(e^N) is > an integer. As far as I know, this remains unresolved. E r \in |Q : e^e^r \in |Q ? Phil -- I find the easiest thing to do is to k/f myself and just troll away -- David Melville on r.a.s.f1
From: Gerry Myerson on 24 Mar 2010 21:43
In article <87y6hhtjcu.fsf(a)kilospaz.fatphil.org>, Phil Carmody <thefatphil_demunged(a)yahoo.co.uk> wrote: > Bart Goddard <goddardbe(a)netscape.net> writes: > > Tony <temptony(a)freemail.hu> wrote: > >> On Mar 14, 1:20�am, bill <b92...(a)yahoo.com> wrote: > >>> On Mar 13, 1:59�pm, Pafnuty Tschebyscheff <th...(a)SDF-EU.ORG> wrote: > >>> > >>> > Is e^e rational? > >>> > Can anyone give me any references to this problem? > >>> > Thanks in advance. > >>> > >>> e^e = 1 + e + e^2/2! + e^3/3! ad infinitum. > >>> > >>> Does this help? > >> > >> No. > > > > An old "West Coast Number Theory" problem, (before 1980, > > I think) asks for the existence of a "Humdrum Number" > > which was defined as an integer N such that e^(e^N) is > > an integer. As far as I know, this remains unresolved. > > E r \in |Q : e^e^r \in |Q ? My hunch is it's no easier (nor harder) for rationals than for integers. -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email) |