Merry Xmas,              Joyeux No\"el,

                      AND            ,             ET ,

             Best wishes for               Meilleurs voeux pour


                         2    0     0    4
                       -x   +x    -y   +y     +z-4/z     > 0



  where

                       z        1
                x  =  e   -   ------
                              z(z+1)

                              z
                       x+y = z


                      yz = x+1


   See also in  http://www.math.ucl.ac.be/~magnus/best04.htm

Very best,  Alphonse Magnus, magnus@anma.ucl.ac.be

                             http://www.math.ucl.ac.be/~magnus/


OK, I wanted a combination of 2 , 0 , 0 , and 4 th powers,
so 2004 close enough to a sum or difference of a square and
a 4th power.  7 at the 4th is 2401 = 2001 + the square of 20, too
good to be ignored. So, I had to add something close to 3.
Three funny relations between 7, 20, and 3 are:
  20 +7 = cube of 3
  20 +1 = 3 times 7
  and I could not resist: 20 is almost the cube of e.

A little more fudging produced the most disgusting 2004 wishes.

The 2005 ones will be much better, I promise.


Script V1.1 session started Mon Dec 22 13:16:30 2003


C:\calc\pari>type wish2004.gp

{
  f(z)=
    x=exp(z)-1/(z*(z+1));y=z^z-x;
   y*z-x-1
}
{
 plot(z=0.05,3.5,f(z));
  z1=3;f1=f(z1);
  z2=3.00001;f2=f(z2);
  for(i=1,5,
     print(z2," ",-x^2+y^4+z2-4/z2);
     z3=z2-f2/(   (f2-f1)/(z2-z1)  );
     z1=z2;f1=f2;z2=z3;f2=f(z3);
     );
  print("x=",x," y=",y," z=",z2," result=",-x^2+y^4+z2-4/z2);
  z1=0.5;f1=f(z1);
  z2=0.50001;f2=f(z2);
  for(i=1,5,
     print(z2," ",-x^2+y^4+z2-4/z2);
     z3=z2-f2/(   (f2-f1)/(z2-z1)  );
     z1=z2;f1=f2;z2=z3;f2=f(z3);
     );
  print("x=",x," y=",y," z=",z2," result=",-x^2+y^4+z2-4/z2);
}

C:\calc\pari>gp-sta

                  GP/PARI CALCULATOR Version 2.1.3 (released)
                i686 running cygwin (ix86 kernel) 32-bit version
              (readline v4.0 enabled, extended help not available)

                       Copyright (C) 2000 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and
comes WITHOUT ANY WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical) support.

   realprecision = 28 significant digits
   seriesprecision = 16 significant terms
   format = g0.28

parisize = 4000000, primelimit = 500000
? \r wish2004

131.00741 |''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''"
          |                                                              :
          |                                                             :|
          |                                                             :|
          |                                                             "|
          |                                                             :|
          |                                                            : |
          |                                                            " |
          |                                                            : |
          |                                                           :  |
          |                                                           "  |
          |                                                              |
          |                                                          "   |
          |                                                              |
          |                                                         "    |
          |                                                        _     |
          x                                                              |
          |                                                       "      |
          |"x__                                                  "       |
          `````""""xxxxxxx_____````````````````````````````````_"`````````
          |                    """""xxxx___                  _"          |
-18.56846 |................................""""xxxx_______xx"............|
          0.05                                                         3.5
3.000009999999999999999999999 2000.049322933633209261497817
3.000091387216367875121574990 2004.062754712875791646812831
3.000091372695862436656719030 2004.062038115832217808253587
3.000091372698169035227578405 2004.062038229664446104878806
3.000091372698169035292956521 2004.062038229664449331333886

x=20.00404338495612718426493938 y=7.001134557467122312807324231 z=3.000091372698169035292956521 result=2004.062038229664449331333886

0.5000099999999999999999999999 -7.575799417654016750682798529
0.3467690322178041859475625497 -7.657638504705500448421988078
0.3718354530744745100573253889 -8.556656078887256348285461933
0.3677247444548175845093260814 -8.471578427902055790355258950
0.3675617942096058676200012128 -8.467794374564010741862735044

x=-0.5451833656203234708755111933 y=1.237384087398233174016221405 z=0.3675630189832248093822627571 result=-8.467822938187760845678167393

? quit
Good bye!

C:\calc\pari>exit

Script completed Mon Dec 22 13:17:13 2003