[OAI-general] math>Number Theory>Fermat's Last Theorem

kerryevans evans kerryme1165@hotmail.com
Sat, 08 Dec 2001 16:25:30 +0000


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<P>The attached is an amateur attempt at proof of&nbsp; Fermat's Last Theorem.&nbsp; It requires only the most basic knowledge of&nbsp; Number Theory. This probably more properly is under the domain of Algebra since it addresses the most&nbsp;fundamental properties of equations.</P></DIV>
<P>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Kerry Evans</P></DIV>
<P>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; kerryme1165.hotmail.com</P>
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<div class=Section1>

<p class=MsoNormal style='text-align:justify;text-indent:.5in'><span
style='mso-tab-count:1'>            </span><span style='mso-tab-count:1'>            </span><span
style="mso-spacerun: yes">          </span>IN DEFENSE OF MR. FERMAT</p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'>During the course of studies on
the Goldbach<span style="mso-spacerun: yes">  </span>Conjecture, using finite
methods, what seems to be an elementary proof of Fermat’s “Last” Theorem has
been found.<span style="mso-spacerun: yes">  </span>Astonishing here is the lucidity
of the arguments and immediacy of their logic.<span style="mso-spacerun: yes"> 
</span>Hopefully, by (numeric) application to the so-called “hard” problems of
Number Theory, some manner of agreement (disputation) will arise.</p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:1'>            </span><span style="mso-spacerun:
yes">           </span>FERMAT’S LAST THEOREM</p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">   </span><span style='mso-tab-count:1'>            </span>Suppose the
following equation has the solution for positive r, a and b.</p>

<p class=MsoNormal style='text-align:justify'><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span
style="mso-spacerun: yes"> </span>(1)<span style="mso-spacerun: yes"> 
</span>r^n = a^n + b^n<span style="mso-spacerun: yes">   </span>where a&gt;0
and b&gt;0.</p>

<p class=MsoBodyText><span style='mso-tab-count:1'>            </span><span
style='mso-tab-count:1'>            </span>Clearly the following congruences
must hold (implied).</p>

<p class=MsoNormal style='text-align:justify'><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span
style="mso-spacerun: yes"> </span>(2)<span style="mso-spacerun: yes"> 
</span>r^n -<span style="mso-spacerun: yes">  </span>b^n = 0<span
style="mso-spacerun: yes">  </span>(mod<span style="mso-spacerun: yes"> 
</span>a^n)</p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes"> </span><span style='mso-tab-count:2'>                       </span><span
style='mso-tab-count:2'>                        </span><span
style="mso-spacerun: yes"> </span>(2)’ r^n – a^n = 0<span style="mso-spacerun:
yes">  </span>(mod<span style="mso-spacerun: yes">  </span>b^n)</p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">     </span></p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify'><span
style="mso-spacerun: yes">    </span>From now on forsaking implicit-only
relations (2) and (2)’ can be used to further specify the consequences of (1),
using the single valued function,F(x,y)</p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><b><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style="mso-spacerun: yes">                  </span>F(x,y)<o:p></o:p></b></p>

<p class=MsoBodyText style='margin-left:15.0pt;text-indent:21.0pt'>Let {c}
represent, i.) c.GE. 0,<span style="mso-spacerun: yes">  </span>the greatest
Integer in c, or ii.) c&lt; 0,. -({|c|} + 1).</p>

<p class=MsoBodyText align=left style='margin-left:15.0pt;text-align:left'><i><span
style="mso-spacerun:
yes">                                                       </span></i>F(x,y)
is Defined<span style="mso-spacerun:
yes">                                                                                     
</span>y: y.NE.0,<span style="mso-spacerun: yes">   </span>F(x,y) = x – ( |y| *
{ x / |y| }),<span style="mso-spacerun: yes">  </span>y: y=0<span
style="mso-spacerun: yes">  </span>F(x,y) is undefined.<span
style="mso-spacerun: yes">  </span>F may be referred to as the “least positive
(or 0) remainder of x on division by |y| function.”</p>

<p class=MsoBodyTextIndent>Explicitly now as [(1) implies (2)] and [(1) implies
(2)’] when [r,a,and b] are positive,<span style="mso-spacerun: yes"> 
</span>(3)<span style="mso-spacerun: yes">   </span>F((r^n – b^n), a^n) =
0<span style="mso-spacerun: yes">   </span>as well as (3)’ F((r^n,a^n), b^n) =
0.</p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span
style="mso-spacerun: yes">                  </span>ORDER</p>

<p class=MsoNormal style='margin-left:15.0pt;text-align:justify;text-indent:
21.0pt'>The following “order” relations are definitive of the assumption [a
&gt; b] when [r, a and b]<span style="mso-spacerun: yes">  </span>are positive.</p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">          </span>(4)<span style="mso-spacerun: yes">  </span>F(r^n,a^n) –
b^n = 0<span style="mso-spacerun: yes">      </span></p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">          </span>(4)’ F(r^n,b^n) – a^n .NE. 0<span style="mso-spacerun:
yes">                                       </span><span style="mso-spacerun:
yes">            </span></p>

<p class=MsoNormal style='margin-top:12.0pt'><span style="mso-spacerun:
yes">                                                                  
</span>PROOF</p>

<p class=MsoNormal style='margin-top:12.0pt;margin-right:0in;margin-bottom:
0in;margin-left:.5in;margin-bottom:.0001pt;line-height:12.0pt'>Let [r,a,and b]
be positive.<span style="mso-spacerun: yes">  </span></p>

<p class=MsoNormal style='margin-left:15.0pt;text-align:justify'><span
style="mso-spacerun: yes">     </span>First suppose a^n = b^n.Then, (1) can be
represented r^n = 2*(a^n) which has no solution in positive in positive
Integers</p>

<p class=MsoNormal style='margin-left:15.0pt;text-align:justify;text-indent:
21.0pt'>Let [a^n &gt; b^n].<span style="mso-spacerun: yes">  </span>Then
F(r^n,a^n) =<span style="mso-spacerun: yes">  </span>b^n<span
style="mso-spacerun: yes">  </span>and (4) immediately follows.<span
style="mso-spacerun: yes">  </span>Regarding<span style="mso-spacerun: yes"> 
</span>(4)’,</p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify'><span
style="mso-spacerun: yes">                                 </span>F(r^n,b^n) =
F(a^n,b^n)<span style="mso-spacerun: yes">  </span>.NE.<span
style="mso-spacerun: yes">  </span>a^n<span style="mso-spacerun: yes">    
</span>so that:<span style="mso-spacerun:
yes">                                                                               
</span></p>

<p class=MsoNormal style='margin-top:12.0pt;text-align:justify'><span
style='font-size:13.0pt;mso-bidi-font-size:12.0pt'><span style="mso-spacerun:
yes">                 </span><span style="mso-spacerun:
yes">              </span>F(r^n,b^n) – a^n = F(r^n,b^n) – F(a^n,b^n) –
d*(b^n)<span style="mso-spacerun: yes">  </span>.NE. 0 <o:p></o:p></span></p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">                                                              
</span>where d .GE.1.</p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun: yes"> 
</span></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='margin-left:15.0pt;text-align:justify;text-indent:
21.0pt'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify;text-indent:.5in'>Conversely,
suppose [a^n is not greater than b^n] while (4) holds.<span
style="mso-spacerun: yes">  </span>Then [b^n &gt; a^n] and transformation
of<span style="mso-spacerun: yes">  </span>a to b<span style="mso-spacerun:
yes">  </span>and<span style="mso-spacerun: yes">  </span>b to a<span
style="mso-spacerun: yes">  </span>has no effect on (1)<span
style="mso-spacerun: yes">  </span>but “rectifies” the representations of (4)
and (4)’ accordingly as [r&gt;<u>b&gt;a</u>&gt;0].<span style="mso-spacerun:
yes">  </span>Thus, (4)’ rightfully becomes, F(r^n,a^n) – b^n<span
style="mso-spacerun: yes">   </span>.NE.<span style="mso-spacerun: yes"> 
</span>0.<b>!</b> which is inconsistent with prior (4).<span
style="mso-spacerun: yes">  </span>Proof of the case (4)’ follows
complementarily. <b>!</b> denotes contradiction. Thus, a^n &gt; b^n may be
defined in this manner.</p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">     </span></p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify'><span
style="mso-spacerun: yes">     </span>Prominent now becomes the anticipation
that (1) has a distinct ordering of a^n and b^n is contradictory.<span
style="mso-spacerun: yes">  </span>A-priori however must be the proof that if
(1) is solvable for some n such that n is greater than or equal to 3, then each
case of n may be “rewritten” (in possibly different<span style="mso-spacerun:
yes">  </span>r, a, b and n) as<span style="mso-spacerun: yes">  </span>v^n’ =
u^n’ + w^n’, where n’ is some divisor of n such that [v&gt;u&gt;w&gt;0] is
necessary.<span style="mso-spacerun: yes">  </span>The different cases of n
will eventually be considered n’ such that n’ is greater than or equal to
3.<span style="mso-spacerun: yes">  </span></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify'><b>OBJECT</b>:
[(1) holds for any n such that n is greater than or equal to 3]<span
style="mso-spacerun: yes">  </span><u>implies</u><span style="mso-spacerun:
yes">  </span>(1) may be rewritten (in possibly different r, a, b and n)
as<span style="mso-spacerun: yes">  </span>v^n’ = u^n’ + w^n’,<span
style="mso-spacerun: yes">  </span>where n’ is greater than or equal to 3 such
that [v, u and w .GT. 0 ] is necessary.</p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='margin-left:15.0pt;text-align:justify;text-indent:
21.0pt'>CASE 1:<span style="mso-spacerun: yes">  </span>n has an odd divisor,
q, greater than 1.</p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify;text-indent:21.0pt'>Let
(1) be written, r^[(n/q)*q] = a^[(n/q)*q] + b^[(n/q)*q] and substituted as<span
style="mso-spacerun: yes">  </span>v^q = u^q + w^q.<span style="mso-spacerun:
yes">  </span>If n/q is even, the fact that<span style="mso-spacerun: yes"> 
</span>plus or minus r, plus or minus a and plus or minus b<span
style="mso-spacerun: yes">  </span>are roots of [v, u, w] implies the latter
are all positive.<span style="mso-spacerun: yes">  </span>On the other hand, if
n/q is odd, original conditions, [a&gt;0 and b&gt;0] imply the relation in
[v,u,w,n’] has strictly positive exponential bases.</p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify;text-indent:21.0pt'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify;text-indent:.5in'>CASE 2:<span
style="mso-spacerun: yes">  </span>n = 2^t where t is greater than or equal to
2.</p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">     </span><span style='mso-tab-count:1'>       </span><span
style='mso-tab-count:1'>            </span>Let(1) be written;</p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify;text-indent:1.75in'>[r^2^(t-1)]^
2 = [a^2^(t-1)]^2<span style="mso-spacerun: yes">  </span>+<span
style="mso-spacerun: yes">  </span>[b^2^(t-1)]^2<span style="mso-spacerun:
yes">     </span>which upon substitution becomes v^2 = u^2 + w^2.<span
style="mso-spacerun: yes">  </span>As before, the fact that 2^(t-1) is even and
the existence of roots plus or minus r, plus or minus a and plus or minus b,
corresponding to [v,u,w] imply they are all positive. Note that the case n=2
cannot be rewritten such that its root is not possibly negative.</p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify;text-indent:21.0pt'>Having
limited the existence of a positive-only solution for a rewrite of (1) such
that [r,a,b,n] goes to [v,u,w,n’]<span style="mso-spacerun: yes">  </span>for
all n greater than or equal to 3, order is specific and consequently
subject<span style="mso-spacerun: yes">  </span>to contradiction.</p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify'>Suppose now that
(1) has been rewritten for some n, where n is equal to or greater than n’ so
that [r,a,b and n] is transformed to a reduced form, redefining [r,a, b and n]
to be [u,v,w and n’]. Thus n’ is greater than or equal to 3 implies
[v&gt;u&gt;w&gt;0].<span style="mso-spacerun: yes">  </span>Consider the
transformation (under n’), T, such that T:v to v, u to w, and w to u.
Respecting (1), T(v,u,w) = Tnot(v,u,w) where Tnot signifies not T.<span
style="mso-spacerun: yes">  </span>It follows that [u&gt;w] can be transformed
to [w&gt;u] such that [u&gt;w] is unaltered.<b>!</b><span style="mso-spacerun:
yes">  </span>This can happen only for the case,u=w, which is moot.
Conclusively, order must be assignable when it exists. ( <b>!</b> denotes
contradiction.).</p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify;text-indent:21.0pt'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">                                                                                                                   
</span></p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify'>SUPPOSE: [r is
conjugated negative in sign to [a]] which implies [(4) and (4)’ are indefinite]
allows a solution to (1) for the case n=2. </p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">                                                           </span></p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">                                                                
</span>EXAMPLE<span style="mso-spacerun:
yes">                                                                                  
</span><span style="mso-spacerun:
yes">                                                                                                          </span></p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify'><span
style="mso-spacerun: yes">     </span>Let r^2 = a^2 + b^2<span
style="mso-spacerun: yes">  </span>(n=2).<span style="mso-spacerun: yes"> 
</span>Divide through by q^2, where q^2 is the greatest (square) divisor common
to [r^2,a^2 and b^2] to obtain the so-called “primitive” form, r^2/q ^2 = a^2/q
^2 + b^2/q ^2.<span style="mso-spacerun: yes">  </span>Rearrange as follows,
conjugating plus or minus |r| with plus or minus |a| where they are of
different sign respectively:</p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">                           </span>|a|^2/q ^2 * [(-|r|)^2/|a|^2 – 1] =
(|a|^2)/q ^2)*[ |r|^2/a^2 – 1] = b^2/q ^2</p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify'><span
style="mso-spacerun: yes">     </span>b^2/q ^2 is substituted by 0, such that
a^2/q ^2, as an Integer multiple of 0, drops out. F is applied such that the
remaining is a relation among squares (preserving n).<span style="mso-spacerun:
yes">  </span>For example,</p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">                            </span>F[(-5)^2/(4)^2 , 9] = F[4^2/4^2 , 9] =
F[5^2/(-4)^2 , 9] =</p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">                            </span>F[(-4)^2/(-4)^2 , 9]</p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify'><span
style="mso-spacerun: yes">     </span>Consequently, F[1^2 , 9] – (-1)^2 =
0<span style="mso-spacerun: yes">  </span>is identifiable as a relation among
squares. Thus the presence of values altering the positivity of [r,a and b] allows
a solution to (1).<span style="mso-spacerun: yes">    </span></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify;text-indent:.5in'>In
the event that [v&gt;u&gt;w&gt;0] (under n’) the latter method clearly cannot
succeed.<span style="mso-spacerun: yes">  </span>In any case, the instances
such that (1) is rewritten (for all non-trivial n, 3 or greater) stand
consistent while example accordingly excepts the case, n=2.</p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal align=center style='text-align:center'>KEY</p>

<p class=MsoNormal align=center style='text-align:center'><b>!</b> denotes
contradiction</p>

<p class=MsoNormal align=center style='text-align:center'>|x| denotes the
absolute value of x</p>

<p class=MsoNormal align=center style='text-align:center'>* denotes
multiplication</p>

<p class=MsoNormal align=center style='text-align:center'>{x} denotes the
greatest Integer in x when x = |x|</p>

<p class=MsoNormal align=center style='text-align:center'>^ denotes
exponentiation</p>

<p class=MsoNormal align=center style='text-align:center'>Tnot denotes “not T”</p>

<p class=MsoNormal align=center style='text-align:center'>.LE. denotes “less
than or equal to”</p>

<p class=MsoNormal align=center style='text-align:center'>.GE. denotes “greater
than or equal to”</p>

<p class=MsoNormal align=center style='text-align:center'>.NE. denotes “not
equal to”</p>

<p class=MsoNormal><span style="mso-spacerun:
yes">                                                                   
</span></p>

<p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal align=center style='text-align:center'><span
style="mso-spacerun: yes"> </span></p>

<p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal align=center style='text-align:center'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'>-</p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='margin-left:.5in;text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><span style="mso-spacerun:
yes">           </span></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal style='text-align:justify'><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span></p>

<p class=MsoNormal style='text-align:justify'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal><span style="mso-spacerun: yes"> </span><span
style='mso-tab-count:2'>                       </span><span style='mso-tab-count:
2'>                        </span><span style="mso-spacerun: yes">  </span></p>

<p class=MsoNormal><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:1'>            </span><span
style="mso-spacerun: yes">     </span><span style="mso-spacerun:
yes">                                                                                                                                                                                                                                                                </span><span
style="mso-spacerun:
yes">                                                               </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:1'>            </span><span
style='mso-tab-count:1'>            </span>-<span style='mso-tab-count:1'>           </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:2'>                        </span><span
style='mso-tab-count:2'>                        </span><span style='mso-tab-count:
2'>                        </span><span style='mso-tab-count:1'>            </span></p>

<p class=MsoNormal><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

<p class=MsoNormal><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>

</div>

</body>

</html>


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