[SPAM] RES: [obm-l] x^2 - xy + y^2 = Cte

["Bouskela" <bouskela@xxxxxxxxx>]

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Ol=E1 Salhab!
=20
Voc=EA est=E1 perto da solu=E7=E3o, entretanto, ainda faltam alguns =
ajustes:
=20
P.ex., se Cte=3D100 , ent=E3o existem apenas 6 solu=E7=F5es para a eq. =
em estudo:
=20
(x, y) =3D (-10, -10) ; (10, 10) ; (10, 0) ; (-10, 0); (0, -10) e (0, =
10)
=20
O mesmo acontece para Cte =3D 1, 3, 4, 9, 12, 16, 25, 27, 36, 48, 64, =
75, 81,
100 etc.
=20
Repare que quando "x" ou "y" s=E3o iguais a "0" seu desenvolvimento =
precisa
ser ajustado...
=20
Sds.,
AB!


  _____ =20

De: owner-obm-l@xxxxxxxxxxxxxx [mailto:owner-obm-l@xxxxxxxxxxxxxx] Em =
nome
de Marcelo Salhab Brogliato
Enviada em: domingo, 29 de junho de 2008 21:16
Para: obm-l@xxxxxxxxxxxxxx
Assunto: Re: [obm-l] x^2 - xy + y^2 =3D Cte


Ol=E1 Bouskela,
suponha que o par (a, b) seja solu=E7=E3o de x^2 - xy + y^2 =3D Cte
ent=E3o, os pares (b, a), (-a, -b), (-b, -a) tamb=E9m s=E3o =
solu=E7=F5es.
Fatorando, temos: (x-y)^2 + xy =3D Cte.
Partindo dela, vemos que os pares (b-a, b) e (a-b, a), pois:
[(a-b)-a]^2 + (a-b)a =3D b^2 + a^2 - ab =3D Cte.
Portanto, provamos que sempre =E9 m=FAltiplo de 6.
Mas, se (b-a, b) e (a-b, a) s=E3o solu=E7=F5es, ent=E3o (b, b-a) e (a, =
a-b) tamb=E9m o
s=E3o.
E, seguindo, temos que (a-b, -b), (-b, a-b), (-a, b-a) e (b-a, -a) =
tamb=E9m o
s=E3o.
Portanto, provamos que sempre =E9 m=FAltiplo de 12.
O que estou errando?

Sobre o n=FAmero finito de solu=E7=F5es, vamos analisar esta equa=E7=E3o =
em fun=E7=E3o de
x, ent=E3o, temos:
x =3D [y +- sqrt(y^2 - 4(y^2-Cte)]/2 =3D [y +- sqrt(4Cte - 3y^2)]/2
isto =E9: 4Cte >=3D 3y^2 .... |y| <=3D sqrt(4Cte/3)
ent=E3o, podemos ir chutando todos os y's poss=EDveis e ir calculando os =
x's.
Mas, para cada y, teremos no m=E1ximo 2 x's.. portanto, temos um =
n=FAmero
limitado de solu=E7=F5es.
Ainda podemos afirmar que o n=FAmero de solu=E7=F5es =E9 <=3D =
2*[2sqrt(4Cte/3) + 1] =3D
4sqrt(4Cte/3) + 2

abra=E7os,
Salhab




2008/6/26 Bouskela <bouskela@xxxxxxxxx>:


Demonstre que a equa=E7=E3o:

x^2 - xy + y^2 =3D Cte

Onde "Cte" =E9 uma constante inteira e positiva.



Tem um n=FAmero FINITO de solu=E7=F5es inteiras; e mais: ESTE N=DAMERO =
=C9 M=DALTIPLO DE
"6".

A depender do valor da constante inteira e positiva "Cte", o n=FAmero de
solu=E7=F5es inteiras desta equa=E7=E3o =E9:

=3D 0 , p.ex.: Cte =3D 2, 5, 6, 8, 10, 11, 14, 15, 17, 18, 20, 22, 23, =
24, 26,
29, 30, 32, 33, 34, 35, 38, 40, 41, 42, 44, 45, 46, 47, 50, 51, 53, 54, =
55,
56, 58, 59, 60, 62, 65, 66, 68, 69, 70, 71, 72, 74, 77, 78, 80, 82, 83, =
85,
86, 87, 88, 89, 90, 92, 94, 95, 96, 98, 99 etc.

=3D 1 , Cte =3D 0

=3D 6 , p.ex.: Cte =3D 1, 3, 4, 9, 12, 16, 25, 27, 36, 48, 64, 75, 81, =
100 etc.

=3D 12 , p.ex.: Cte =3D 7, 13, 19, 21, 28, 31, 37, 39, 43, 52, 57, 61, =
63, 67,
73, 76, 79, 84, 93, 97 etc.

=3D 18 , p.ex.: Cte =3D 49 etc.

=3D 24 , p.ex.: Cte =3D 91 etc.

Sds.,

AB



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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=3DContent-Type content=3D"text/html; =
charset=3Diso-8859-1">
<META content=3D"MSHTML 6.00.6000.16674" name=3DGENERATOR></HEAD>
<BODY>
<DIV dir=3Dltr align=3Dleft><FONT face=3DVerdana size=3D2><SPAN=20
class=3D015405801-30062008>Ol=E1 Salhab!</SPAN></FONT></DIV>
<DIV dir=3Dltr align=3Dleft><FONT face=3DVerdana size=3D2><SPAN=20
class=3D015405801-30062008></SPAN></FONT>&nbsp;</DIV>
<DIV dir=3Dltr align=3Dleft><FONT face=3DVerdana size=3D2><SPAN=20
class=3D015405801-30062008>Voc=EA est=E1 perto da solu=E7=E3o, =
entretanto, ainda faltam=20
alguns ajustes:</SPAN></FONT></DIV>
<DIV dir=3Dltr align=3Dleft><FONT face=3DVerdana size=3D2><SPAN=20
class=3D015405801-30062008></SPAN></FONT>&nbsp;</DIV>
<DIV dir=3Dltr align=3Dleft><FONT face=3DVerdana size=3D2><SPAN=20
class=3D015405801-30062008>P.ex., se Cte=3D100 , ent=E3o existem apenas =
6 solu=E7=F5es=20
para a eq. em estudo:</SPAN></FONT></DIV>
<DIV dir=3Dltr align=3Dleft><FONT face=3DVerdana size=3D2><SPAN=20
class=3D015405801-30062008></SPAN></FONT>&nbsp;</DIV>
<DIV dir=3Dltr align=3Dleft><FONT face=3DVerdana size=3D2><SPAN=20
class=3D015405801-30062008>(x, y) =3D (-10, -10) ; (10, 10) ; (10, 0) ; =
(-10, 0);=20
(0, -10) e (0, 10)</SPAN></FONT></DIV>
<DIV dir=3Dltr align=3Dleft><FONT face=3DVerdana size=3D2><SPAN=20
class=3D015405801-30062008></SPAN></FONT>&nbsp;</DIV>
<DIV dir=3Dltr align=3Dleft><FONT><SPAN =
class=3D015405801-30062008></SPAN></FONT><SPAN=20
class=3D015405801-30062008><FONT face=3DVerdana size=3D2>O mesmo =
acontece para Cte =3D=20
1, 3, 4, 9, 12, 16, 25, 27, 36, 48, 64, 75, 81, 100 =
etc.</FONT></SPAN></DIV>
<DIV align=3Dleft><FONT face=3DVerdana size=3D2></FONT>&nbsp;</DIV>
<DIV align=3Dleft><FONT face=3DVerdana size=3D2><SPAN =
class=3D015405801-30062008>Repare=20
que quando "x" ou "y" s=E3o iguais a "0" seu =
desenvolvimento&nbsp;precisa ser=20
ajustado...</SPAN></FONT></DIV>
<DIV align=3Dleft><FONT face=3DVerdana size=3D2></FONT>&nbsp;</DIV>
<DIV><SPAN class=3D015405801-30062008><FONT face=3DVerdana=20
size=3D2>Sds.,</FONT></SPAN></DIV>
<DIV><SPAN class=3D015405801-30062008><FONT face=3DVerdana=20
size=3D2>AB!</FONT></SPAN></DIV><BR>
<BLOCKQUOTE=20
style=3D"PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #000000 2px =
solid; MARGIN-RIGHT: 0px">
  <DIV class=3DOutlookMessageHeader lang=3Dpt-br dir=3Dltr align=3Dleft>
  <HR tabIndex=3D-1>
  <FONT face=3DTahoma size=3D2><B>De:</B> owner-obm-l@xxxxxxxxxxxxxx=20
  [mailto:owner-obm-l@xxxxxxxxxxxxxx] <B>Em nome de </B>Marcelo Salhab=20
  Brogliato<BR><B>Enviada em:</B> domingo, 29 de junho de 2008=20
  21:16<BR><B>Para:</B> obm-l@xxxxxxxxxxxxxx<BR><B>Assunto:</B> Re: =
[obm-l] x^2=20
  - xy + y^2 =3D Cte<BR></FONT><BR></DIV>
  <DIV></DIV>Ol=E1 Bouskela,<BR>suponha que o par (a, b) seja =
solu=E7=E3o de x^2 - xy=20
  + y^2 =3D Cte<BR>ent=E3o, os pares (b, a), (-a, -b), (-b, -a) tamb=E9m =
s=E3o=20
  solu=E7=F5es.<BR>Fatorando, temos: (x-y)^2 + xy =3D Cte.<BR>Partindo =
dela, vemos que=20
  os pares (b-a, b) e (a-b, a), pois:<BR>[(a-b)-a]^2 + (a-b)a =3D b^2 + =
a^2 - ab =3D=20
  Cte.<BR>Portanto, provamos que sempre =E9 m=FAltiplo de 6.<BR>Mas, se =
(b-a, b) e=20
  (a-b, a) s=E3o solu=E7=F5es, ent=E3o (b, b-a) e (a, a-b) tamb=E9m o =
s=E3o.<BR>E, seguindo,=20
  temos que (a-b, -b), (-b, a-b), (-a, b-a) e (b-a, -a) tamb=E9m o=20
  s=E3o.<BR>Portanto, provamos que sempre =E9 m=FAltiplo de 12.<BR>O que =
estou=20
  errando?<BR><BR>Sobre o n=FAmero finito de solu=E7=F5es, vamos =
analisar esta equa=E7=E3o=20
  em fun=E7=E3o de x, ent=E3o, temos:<BR>x =3D [y +- sqrt(y^2 - =
4(y^2-Cte)]/2 =3D [y +-=20
  sqrt(4Cte - 3y^2)]/2<BR>isto =E9: 4Cte &gt;=3D 3y^2 .... |y| &lt;=3D=20
  sqrt(4Cte/3)<BR>ent=E3o, podemos ir chutando todos os y's poss=EDveis =
e ir=20
  calculando os x's.<BR>Mas, para cada y, teremos no m=E1ximo 2 x's.. =
portanto,=20
  temos um n=FAmero limitado de solu=E7=F5es.<BR>Ainda podemos afirmar =
que o n=FAmero de=20
  solu=E7=F5es =E9 &lt;=3D 2*[2sqrt(4Cte/3) + 1] =3D 4sqrt(4Cte/3) +=20
  2<BR><BR>abra=E7os,<BR>Salhab<BR><BR><BR><BR>
  <DIV class=3Dgmail_quote>2008/6/26 Bouskela &lt;<A=20
  href=3D"mailto:bouskela@xxxxxxxxx";>bouskela@xxxxxxxxx</A>&gt;:<BR>
  <BLOCKQUOTE class=3Dgmail_quote=20
  style=3D"PADDING-LEFT: 1ex; MARGIN: 0pt 0pt 0pt 0.8ex; BORDER-LEFT: =
rgb(204,204,204) 1px solid">
    <DIV>
    <DIV>
    <P><FONT face=3DVerdana size=3D2>Demonstre que a =
equa=E7=E3o:</FONT></P>
    <P><FONT face=3DVerdana size=3D2>x^2 - xy + y^2 =3D Cte</FONT></P>
    <P><FONT face=3DVerdana size=3D2>Onde "Cte" =E9 uma constante =
inteira e=20
    positiva.</FONT></P>
    <P><FONT face=3DVerdana size=3D2></FONT></P>
    <P><FONT face=3DVerdana size=3D2>Tem um n=FAmero FINITO de =
solu=E7=F5es inteiras; e=20
    mais: ESTE N=DAMERO =C9 M=DALTIPLO DE "6".</FONT></P>
    <P><FONT face=3DVerdana size=3D2>A depender do valor da constante =
inteira e=20
    positiva "Cte", o n=FAmero de solu=E7=F5es inteiras desta =
equa=E7=E3o =E9:</FONT></P>
    <P><FONT face=3DVerdana size=3D2>=3D 0 , p.ex.: Cte =3D 2, 5, 6, 8, =
10, 11, 14, 15,=20
    17, 18, 20, 22, 23, 24, 26, 29, 30, 32, 33, 34, 35, 38, 40, 41, 42, =
44, 45,=20
    46, 47, 50, 51, 53, 54, 55, 56, 58, 59, 60, 62, 65, 66, 68, 69, 70, =
71, 72,=20
    74, 77, 78, 80, 82, 83, 85, 86, 87, 88, 89, 90, 92, 94, 95, 96, 98, =
99=20
    etc.</FONT></P>
    <P><FONT face=3DVerdana size=3D2>=3D 1 , Cte =3D 0</FONT></P>
    <P><FONT face=3DVerdana size=3D2>=3D 6 , p.ex.: Cte =3D 1, 3, 4, 9, =
12, 16, 25, 27,=20
    36, 48, 64, 75, 81, 100 etc.</FONT></P>
    <P><FONT face=3DVerdana size=3D2>=3D 12 , p.ex.: Cte =3D 7, 13, 19, =
21, 28, 31, 37,=20
    39, 43, 52, 57, 61, 63, 67, 73, 76, 79, 84, 93, 97 etc.</FONT></P>
    <P><FONT face=3DVerdana size=3D2>=3D 18 , p.ex.: Cte =3D 49 =
etc.</FONT></P>
    <P><FONT face=3DVerdana size=3D2>=3D 24 , p.ex.: Cte =3D 91 =
etc.</FONT></P>
    <P><FONT face=3DVerdana size=3D2>Sds.,</FONT></P>
    <P><SPAN><FONT face=3DVerdana=20
  =
size=3D2>AB</FONT></SPAN></P></DIV></DIV></BLOCKQUOTE></DIV><BR></BLOCKQU=
OTE></BODY></HTML>

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