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[SPAM] Re: [obm-l] 2^k=k^2
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Vejam o Livro "Meu Professor de Matem=E1tica" escrito pelo Elon.
Tem um cap=EDtulo inteiro falando sobre ela...
Abra=E7os.
Em 03/11/07, Nicolau C. Saldanha <nicolau@xxxxxxxxxxxxxx> escreveu:
>
> On Nov 2, 2007 1:55 PM, Carlos Nehab <nehab@xxxxxxxxxxxxxxx> wrote:
> >
> > Oi, Lenadro,
> >
> > D=EA uma olhada em
> > http://www.mat.puc-rio.br/~obmlistas/obm-l.200611/msg00093.html
>
> Complementando...
>
> A solu=E7=E3o negativa x =E9 transcendente.
>
> De fato, x obviamente n=E3o =E9 inteiro.
>
> Se ela fosse racional n=E3o inteira x =3D p/q ent=E3o y =3D 2^x n=E3o ser=
ia racional
> pois seria raiz de y^q - 2^p =3D 0 que n=E3o admite raiz racional n=E3o i=
nteira.
> Mas isso =E9 um absurdo pois x racional implica y =3D x^2 racional.
> Assim x n=E3o =E9 racional.
>
> Se x fosse alg=E9brico irracional ent=E3o y =3D 2^x seria transcendente
> pois alg=E9brico diferente de 0 e 1 elevado a alg=E9brico irracional d=E1
> transcendente.
> Por outro lado y =3D x^2 =E9 alg=E9brico, absurdo.
>
> Claro que isso n=E3o exclui em princ=EDpio a possibilidade de x admitir
> uma express=E3o
> relativamente simples em termos de fun=E7=F5es elementares (exp, log, sin=
,
> cos, ...)
> mas acho extremamente improv=E1vel que isto ocorra.
>
> N.
>
> =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
> Instru=E7=F5es para entrar na lista, sair da lista e usar a lista em
> http://www.mat.puc-rio.br/~obmlistas/obm-l.html
> =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
>
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Vejam o Livro "Meu Professor de Matem=E1tica" escrito pelo Elon.<=
br>Tem um cap=EDtulo inteiro falando sobre ela...<br>Abra=E7os.<br><br><div=
><span class=3D"gmail_quote">Em 03/11/07, <b class=3D"gmail_sendername">Nic=
olau C. Saldanha
</b> <<a href=3D"mailto:nicolau@xxxxxxxxxxxxxx";>nicolau@xxxxxxxxxxxxxx</=
a>> escreveu:</span><blockquote class=3D"gmail_quote" style=3D"border-le=
ft: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: =
1ex;">
On Nov 2, 2007 1:55 PM, Carlos Nehab <<a href=3D"mailto:nehab@xxxxxxxxxx=
om.br">nehab@xxxxxxxxxxxxxxx</a>> wrote:<br>><br>> Oi, =
Lenadro,<br>><br>> D=EA uma olhada em<br>> <=
a href=3D"http://www.mat.puc-rio.br/~obmlistas/obm-l.200611/msg00093.html";>
http://www.mat.puc-rio.br/~obmlistas/obm-l.200611/msg00093.html</a><br><br>=
Complementando...<br><br>A solu=E7=E3o negativa x =E9 transcendente.<br><br=
>De fato, x obviamente n=E3o =E9 inteiro.<br><br>Se ela fosse racional n=E3=
o inteira x =3D p/q ent=E3o y =3D 2^x n=E3o seria racional
<br>pois seria raiz de y^q - 2^p =3D 0 que n=E3o admite raiz racional n=E3o=
inteira.<br>Mas isso =E9 um absurdo pois x racional implica y =3D x^2 raci=
onal.<br>Assim x n=E3o =E9 racional.<br><br>Se x fosse alg=E9brico irracion=
al ent=E3o y =3D 2^x seria transcendente
<br>pois alg=E9brico diferente de 0 e 1 elevado a alg=E9brico irracional d=
=E1<br>transcendente.<br>Por outro lado y =3D x^2 =E9 alg=E9brico, absurdo.=
<br><br>Claro que isso n=E3o exclui em princ=EDpio a possibilidade de x adm=
itir<br>uma express=E3o
<br>relativamente simples em termos de fun=E7=F5es elementares (exp, log, s=
in, cos, ...)<br>mas acho extremamente improv=E1vel que isto ocorra.<br><br=
>N.<br><br>=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D
<br>Instru=E7=F5es para entrar na lista, sair da lista e usar a lista em<br=
><a href=3D"http://www.mat.puc-rio.br/~obmlistas/obm-l.html";>http://www.mat=
.puc-rio.br/~obmlistas/obm-l.html</a><br>=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
<br></blockquote></div><br>
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