Mathematica 2.0 for SPARC
Copyright 1988-91 Wolfram Research, Inc.
 -- X11 windows graphics initialized -- 

In[1]:= <<Euclid.m
Package Euclid` loaded
0.166667 Second

In[2]:= (* Demonstration of the generic Euclidean algorithm. *)\
Euclid`ETrace[s__]:=Print[s]
0. Second

In[3]:= ExtendedGeneric[a_, b_, quotient_, unitnormalize_]  returns {g, s, t} such that
   g == GCD[a, b] and s a + t b = g,  where quotient must be bound to a quotient
   function on the type  of elements in a and b and unitnormalize must be a
   function   for normalizing the GCD to g by multipying by a unit.
 Examples:	 ExtendedGeneric[120, 75, Quotient, Abs];
 			 ExtendedGeneric[120, 75 + 5 I,
 				 Gaussian`Quotient, Gaussian`Normalize];
 			 ExtendedGeneric[x^6-1, x^4-1,
 				 Function[PolynomialQuotient[#1, #2, x] ], 				
   Function[Polynom`MakeMonic[#, x] ] ].

In[3]:=                                   (* The first instantiation is on the integers, with standard
division; unit-normalization is by taking absolute values. *)\
gcd1[x_, y_]:=ExtendedGeneric[x, y, System`Quotient, Abs]
0. Second

In[4]:= a = Prime[100] Prime[1000]
0.0166667 Second

Out[4]= 4284179

In[5]:= b = Prime[100] Prime[1001]
0. Second

Out[5]= 4288507

In[6]:= gcd1[a, b]
Dividend=4284179, Divisor=4288507, sy=0
Dividend=4288507, Divisor=4284179, sy=1
Dividend=4284179, Divisor=4328, sy=-1
Dividend=4328, Divisor=3787, sy=990
Dividend=3787, Divisor=541, sy=-991
0.05 Second

Out[6]= {541, -991, 990}

In[7]:= (* There is an alternate instantiation on the integers, with
division that leaves the absolutely smallest remainder. *)\
<<Numbers.m

Quotient::shdw: Warning: Symbol Quotient appears in multiple contexts 
    {Numbers`, System`}; definitions in context Numbers`
     may shadow or be shadowed by other definitions.
Package Numbers` loaded
0.5 Second

In[8]:= Numbers`Quotient[x_Integer, y_Integer] returns the quotient that leaves  the
   absolutely smallest remainder.

In[8]:=                                     (* E.g., the alternate remainder function computes *)\
Numbers`Remainder[10, 6]
0. Second

Out[8]= -2

In[9]:= gcd2[x_, y_]:=ExtendedGeneric[x, y, Numbers`Quotient, Abs]
0. Second

In[10]:= (* Same test inputs as before. Observe the savings in the 
number of divisions. *)\
gcd2[a, b]
Dividend=4284179, Divisor=4288507, sy=0
Dividend=4288507, Divisor=4328, sy=-1
Dividend=4328, Divisor=541, sy=-991
0.05 Second

Out[10]= {541, -991, 990}

In[11]:= (* The next example does a Gaussian integer GCD *)\
<<Gaussian.m

Quotient::shdw: Warning: Symbol Quotient appears in multiple contexts 
    {Gaussian`, System`, Numbers`}; definitions in context Gaussian`
     may shadow or be shadowed by other definitions.

Remainder::shdw: 
   Warning: Symbol Remainder appears in multiple contexts {Gaussian`, Numbers`}
    ; definitions in context Gaussian`
     may shadow or be shadowed by other definitions.
0.1 Second

In[12]:= Quotient[a_, b_] computes a Gaussian integral quotient of two Gaussian integers.

In[12]:=                                       
0. Second

In[13]:= Normalize[a_] returns b such that b/a = 1, -1, I, or -I and -Pi/4 < Arg[b] <=
   Pi/4

In[13]:=                                         gcd3[x_, y_]:=ExtendedGeneric[x, y, Gaussian`Quotient, Gaussian`Normalize]
0. Second

In[14]:= g1 = (10 + 5 I)(7 - 23 I)
0.0166667 Second

Out[14]= 185 - 195 I

In[15]:= g2 = (7 - 23 I)(13 - 11 I)
0. Second

Out[15]= -162 - 376 I

In[16]:= gcd3[g1, g2]
Dividend=185 - 195 I, Divisor=-162 - 376 I, sy=0
Dividend=-162 - 376 I, Divisor=191 + 33 I, sy=-1
Dividend=191 + 33 I, Divisor=39 + 37 I, sy=-2 + I
0.1 Second

Out[16]= {39 + 37 I, -2 + I, 1 + I}

In[17]:= (* The next examples do polynomial GCDs *)\
<<Polynom.m

Global`Modulus::shdw: 
   Warning: Symbol Modulus appears in multiple contexts {Global`, System`}
    ; definitions in context Global`
     may shadow or be shadowed by other definitions.
Package Polynom` loaded
0.316667 Second

In[18]:= f = RandomPoly[10, 10, x]
0.0166667 Second

                       2    3      4      5      6       7    8      9      10
Out[18]= -6 - 8 x - 6 x  + x  + 6 x  - 8 x  + 7 x  + 10 x  - x  - 4 x  - 7 x

In[19]:= g = RandomPoly[9, 10, x]
0.0166667 Second

                      2      3      4      5      6    7    8       9
Out[19]= 7 + 7 x + 7 x  + 9 x  - 9 x  - 5 x  + 3 x  - x  + x  - 10 x

In[20]:= qr2[a_, b_] := PolynomialQuotient[a, b, x]
0. Second

In[21]:= ExtendedGeneric[f, g, qr2, Function[#]]
                       2    3      4      5      6       7    8      9      10
Dividend=-6 - 8 x - 6 x  + x  + 6 x  - 8 x  + 7 x  + 10 x  - x  - 4 x  - 7 x
 
                           2      3      4      5      6    7    8       9
>   , Divisor=7 + 7 x + 7 x  + 9 x  - 9 x  - 5 x  + 3 x  - x  + x  - 10 x , sy=0
                      2      3      4      5      6    7    8       9
Dividend=7 + 7 x + 7 x  + 9 x  - 9 x  - 5 x  + 3 x  - x  + x  - 10 x , Divisor=
 
                            2        3        4       5        6        7
      929    1619 x   1419 x    813 x    393 x    13 x    909 x    837 x
>   -(---) - ------ - ------- - ------ + ------ + ----- + ------ + ------ - 
      100     100       100      100      100      20      100      100
 
         8
     77 x
>    -----, sy=1
      100
                                 2        3        4       5        6
           929    1619 x   1419 x    813 x    393 x    13 x    909 x
Dividend=-(---) - ------ - ------- - ------ + ------ + ----- + ------ + 
           100     100       100      100      100      20      100
 
          7       8                                         2            3
     837 x    77 x           7745700   14183200 x   107900 x    7888200 x
>    ------ - -----, Divisor=------- + ---------- + --------- + ---------- - 
      100      100            5929        5929         49          5929
 
              4           5            6            7
     2686500 x    871300 x    7570600 x    7647100 x        829300    1000 x
>    ---------- - --------- - ---------- - ----------, sy=-(------) - ------
        5929        5929         5929         5929           5929       77
                                        2            3            4
         7745700   14183200 x   107900 x    7888200 x    2686500 x
Dividend=------- + ---------- + --------- + ---------- - ---------- - 
          5929        5929         49          5929         5929
 
             5            6            7
     871300 x    7570600 x    7647100 x
>    --------- - ---------- - ----------, Divisor=
       5929         5929         5929
 
                                                2               3
       5840960279     375536931 x   3881965318 x    3566993122 x
>   -(------------) - ----------- - ------------- - ------------- - 
      146195346025    11695627682   146195346025    146195346025
 
                  4                5                6
     21210268233 x    70181294337 x    39997075503 x
>    -------------- - -------------- + --------------, sy=
      292390692050     584781384100     292390692050
 
                                        2
     5634844523    4942005299 x   5929 x
>   ------------ - ------------ + -------
    584781384100   584781384100   764710
                                                     2               3
            5840960279     375536931 x   3881965318 x    3566993122 x
Dividend=-(------------) - ----------- - ------------- - ------------- - 
           146195346025    11695627682   146195346025    146195346025
 
                  4                5                6
     21210268233 x    70181294337 x    39997075503 x
>    -------------- - -------------- + --------------, Divisor=
      292390692050     584781384100     292390692050
 
                                                                           2
    54231954232405053200   18632287793924753150 x   128775091039686201400 x
>   -------------------- + ---------------------- + ------------------------ + 
     89940183774255507       12848597682036501         89940183774255507
 
                           3                        4                         5
     58506610246029060800 x    9279670351059329450 x    88258778771652120875 x
>    ----------------------- - ---------------------- - -----------------------
        89940183774255507         4733693882855553         29980061258085169
 
                                                                            2
         2678445957891447175   2126377829199385275 x   5110868247287491300 x
>   , sy=------------------- - --------------------- + ---------------------- + 
          89940183774255507      29980061258085169       89940183774255507
 
                    3
     2923906920500 x
>    ----------------
       39997075503
         54231954232405053200   18632287793924753150 x
Dividend=-------------------- + ---------------------- + 
          89940183774255507       12848597682036501
 
                            2                         3
     128775091039686201400 x    58506610246029060800 x
>    ------------------------ + ----------------------- - 
        89940183774255507          89940183774255507
 
                          4                         5
     9279670351059329450 x    88258778771652120875 x
>    ---------------------- - -----------------------, Divisor=
        4733693882855553         29980061258085169
 
      13297902229888107359553498473     17275917939437498893011136292 x
>   -(------------------------------) - ------------------------------- - 
      159846648516390712948180741875    159846648516390712948180741875
 
                                   2                                3
     9885115331640005457294023191 x    721782918320087886849693722 x
>    ------------------------------- - ------------------------------ + 
     159846648516390712948180741875    159846648516390712948180741875
 
                                    4
     15706326437806793277658343867 x
>    --------------------------------, sy=
      159846648516390712948180741875
 
     1198901036243889120682374758    105564728339176734059940739 x
>   ------------------------------ - ----------------------------- + 
    159846648516390712948180741875   53282216172130237649393580625
 
                                 2                                3
     61178730070804081773083123 x     16634449370813855608824797 x
>    ------------------------------ - ----------------------------- + 
     159846648516390712948180741875   6393865940655628517927229675
 
                        4
     59960122516170338 x
>    --------------------
     17651755754330424175
           13297902229888107359553498473     17275917939437498893011136292 x
Dividend=-(------------------------------) - ------------------------------- - 
           159846648516390712948180741875    159846648516390712948180741875
 
                                   2                                3
     9885115331640005457294023191 x    721782918320087886849693722 x
>    ------------------------------- - ------------------------------ + 
     159846648516390712948180741875    159846648516390712948180741875
 
                                    4
     15706326437806793277658343867 x
>    --------------------------------, Divisor=
      159846648516390712948180741875
 
      9638015800779912366487272118099771460817500
>   -(-------------------------------------------) - 
       8228425153881912905325465236460620335081
 
     1620228779273372179060604926770922145506875 x
>    --------------------------------------------- - 
        484025009051877229725027366850624725593
 
                                                  2
     3673706749814508541451943479243514997270000 x
>    ---------------------------------------------- - 
        1175489307697416129332209319494374333583
 
                                                   3
     10685562608494990507870892787734541589594375 x
>    -----------------------------------------------, sy=
        8228425153881912905325465236460620335081
 
    1561301177875831317196930647259096387338125
>   ------------------------------------------- + 
     8228425153881912905325465236460620335081
 
     917752728752155441379665468995273763174375 x
>    -------------------------------------------- + 
       8228425153881912905325465236460620335081
 
                                                2
     46313173760702305035323713776795769249375 x
>    -------------------------------------------- + 
       8228425153881912905325465236460620335081
 
                                                 3
     239310310978610605406060863092005313046875 x
>    --------------------------------------------- - 
       8228425153881912905325465236460620335081
 
                                                4
     45258674604055890498028066196825163613125 x
>    -------------------------------------------- + 
       8228425153881912905325465236460620335081
 
                                      5
     1598466485163907129481807418750 x
>    ----------------------------------
       15706326437806793277658343867
           9638015800779912366487272118099771460817500
Dividend=-(-------------------------------------------) - 
            8228425153881912905325465236460620335081
 
     1620228779273372179060604926770922145506875 x
>    --------------------------------------------- - 
        484025009051877229725027366850624725593
 
                                                  2
     3673706749814508541451943479243514997270000 x
>    ---------------------------------------------- - 
        1175489307697416129332209319494374333583
 
                                                   3
     10685562608494990507870892787734541589594375 x
>    -----------------------------------------------, Divisor=
        8228425153881912905325465236460620335081
 
    287518189154650082426352957906975037030979159962751062629
>   ---------------------------------------------------------- + 
    2142952310602084854981094368499182615059401373068228230625
 
     909635968022376178487653345359476798284031392367386944134 x
>    ----------------------------------------------------------- + 
     2142952310602084854981094368499182615059401373068228230625
 
                                                               2
     22701354424652195485798404016587713758831295876054037518 x
>    -----------------------------------------------------------, sy=
      85718092424083394199243774739967304602376054922729129225
 
       59383005598980303483966508498970538660411565759783713464
>   -(----------------------------------------------------------) - 
      2142952310602084854981094368499182615059401373068228230625
 
     3566693476936741254928575588167240911305216526467818966 x
>    --------------------------------------------------------- + 
     428590462120416970996218873699836523011880274613645646125
 
                                                               2
     16666643602110884814769099699300419665730984277199761052 x
>    ----------------------------------------------------------- - 
     2142952310602084854981094368499182615059401373068228230625
 
                                                               3
     16228122822649408772070771163720530927576367149170540204 x
>    ----------------------------------------------------------- + 
     2142952310602084854981094368499182615059401373068228230625
 
                                                              4
     2836454325760430722560973166313993531205168576297420137 x
>    ---------------------------------------------------------- - 
     428590462120416970996218873699836523011880274613645646125
 
                                                               5
     41363572735082217136145786924887437773141731790060792399 x
>    ----------------------------------------------------------- + 
     2142952310602084854981094368499182615059401373068228230625
 
                                                6
     16456850307763825810650930472921240670162 x
>    --------------------------------------------
     2137112521698998101574178557546908317918875
Dividend=287518189154650082426352957906975037030979159962751062629 / 
 
>     2142952310602084854981094368499182615059401373068228230625 + 
 
>    (909635968022376178487653345359476798284031392367386944134 x) / 
 
>     2142952310602084854981094368499182615059401373068228230625 + 
 
                                                                2
>    (22701354424652195485798404016587713758831295876054037518 x ) / 
 
>     85718092424083394199243774739967304602376054922729129225, Divisor=
 
>   -(201195100173316601032961836661071203588516435156286358662054851545339433\
 
>        13 / 31315317516775925410504346825730029780819357133011861184328347146\
 
>        730002) - (31830198869621807232882319747387470058197806325670685580982\
 
>        581190530394023 x) / 
 
>     31315317516775925410504346825730029780819357133011861184328347146730002
 
>   , sy=1260800845210683322744572348631429735236087154060258814481632564737297\
 
>      529 / 156576587583879627052521734128650148904096785665059305921641735733\
 
>      65001 - (127822152989804386980044793351349218997174869301581983216579833\
 
>        701450510 x) / 
 
>     2236808394055423243607453344695002127201382652357990084594881939052143 - 
 
>    (70819981814445524418002248406849032617627822676142988389408201613475222 
 
         2
>       x ) / 15657658758387962705252173412865014890409678566505930592164173573\
 
>      365001 + (83592876808441330252666234642473776321614777800744359292605541\
 
                    3
>        304509217 x ) / 
 
>     2236808394055423243607453344695002127201382652357990084594881939052143 - 
 
>    (47109316643557533991766134349069537484321268932303391322296187887912645 
 
         4
>       x ) / 28468470469796295827731224387027299800744870120919873803934861042\
 
>      48182 + (910406808195805777028207770882213288675832038551826400309117021\
 
                    5
>        427628314 x ) / 
 
>     15657658758387962705252173412865014890409678566505930592164173573365001 - 
 
>    (2013436360466697899465526344777765478513744115668806353364428950625071605 
 
         6
>       x ) / 31315317516775925410504346825730029780819357133011861184328347146\
 
>      730002 + (428590462120416970996218873699836523011880274613645646125 
 
         7
>       x ) / 11350677212326097742899202008293856879415647938027018759
Dividend=-(20119510017331660103296183666107120358851643515628635866205485154533\
 
>        943313 / 
 
>       31315317516775925410504346825730029780819357133011861184328347146730002)
 
>      - (318301988696218072328823197473874700581978063256706855809825811905303\
 
>        94023 x) / 
 
>     31315317516775925410504346825730029780819357133011861184328347146730002
 
>   , Divisor=-(837023808396956065226676784536597415767597680953014347183557736\
 
>       4072411217256273265305740711 / 
 
>      295492331731770737179004968447431561595972282836347207013486677347543009\
 
>       258394021449002765441), sy=
 
>   -(2170413037111061516094164568447105732142428450707398240390365473985744941\
 
>        726354272937427872 / 
 
>       29549233173177073717900496844743156159597228283634720701348667734754300\
 
>        9258394021449002765441) - 
 
>    (1767438782064252015314047240496288752313730049695252449442695491800171239\
 
>        51099306772565022 x) / 
 
>     9849744391059024572633498948247718719865742761211573567116222578251433641\
 
>      9464673816334255147 - (2439534319785691485010207925423721242828406634381\
 
                                                     2
>        546957415861433005031003688724415920405180 x ) / 
 
>     2954923317317707371790049684474315615959722828363472070134866773475430092\
 
>      58394021449002765441 + (207044244667491240962410736313492992164910725164\
 
                                                     3
>        171881852705114218586328735348495887611940 x ) / 
 
>     2954923317317707371790049684474315615959722828363472070134866773475430092\
 
>      58394021449002765441 + (359617048444991504151859371514186636746071255086\
 
                                                      4
>        4587173677003475033965765395277373582714421 x ) / 
 
>     2954923317317707371790049684474315615959722828363472070134866773475430092\
 
>      58394021449002765441 - (292469002630273960775712324413620077597129239489\
 
                                                     5
>        088833841479823159559862182687619017881045 x ) / 
 
>     3283248130353008190877832982749239573288580920403857855705407526083811213\
 
>      9821557938778085049 + (1946909658270247557586716092101390926380407572537\
 
                                                     6
>        356125589174952353647922397234607605735778 x ) / 
 
>     2954923317317707371790049684474315615959722828363472070134866773475430092\
 
>      58394021449002765441 - (212827333839531279408223988950204242674016326589\
 
                                                      7
>        6952213919734289338967670417476976749752274 x ) / 
 
>     2954923317317707371790049684474315615959722828363472070134866773475430092\
 
>      58394021449002765441 + (313153175167759254105043468257300297808193571330\
 
                                   8
>        118611843283471467300020 x ) / 
 
>     31830198869621807232882319747387470058197806325670685580982581190530394023
11.9167 Second

Out[21]= {-(8370238083969560652266767845365974157675976809530143471835577364072\
 
>        411217256273265305740711 / 
 
>       29549233173177073717900496844743156159597228283634720701348667734754300\
 
>        9258394021449002765441), 
 
>    -(217041303711106151609416456844710573214242845070739824039036547398574494\
 
>         1726354272937427872 / 
 
>        2954923317317707371790049684474315615959722828363472070134866773475430\
 
>         09258394021449002765441) - 
 
>     (176743878206425201531404724049628875231373004969525244944269549180017123\
 
>         951099306772565022 x) / 
 
>      984974439105902457263349894824771871986574276121157356711622257825143364\
 
>       19464673816334255147 - (24395343197856914850102079254237212428284066343\
 
                                                        2
>         81546957415861433005031003688724415920405180 x ) / 
 
>      295492331731770737179004968447431561595972282836347207013486677347543009\
 
>       258394021449002765441 + (2070442446674912409624107363134929921649107251\
 
                                                        3
>         64171881852705114218586328735348495887611940 x ) / 
 
>      295492331731770737179004968447431561595972282836347207013486677347543009\
 
>       258394021449002765441 + (3596170484449915041518593715141866367460712550\
 
                                                         4
>         864587173677003475033965765395277373582714421 x ) / 
 
>      295492331731770737179004968447431561595972282836347207013486677347543009\
 
>       258394021449002765441 - (2924690026302739607757123244136200775971292394\
 
                                                        5
>         89088833841479823159559862182687619017881045 x ) / 
 
>      328324813035300819087783298274923957328858092040385785570540752608381121\
 
>       39821557938778085049 + (19469096582702475575867160921013909263804075725\
 
                                                        6
>         37356125589174952353647922397234607605735778 x ) / 
 
>      295492331731770737179004968447431561595972282836347207013486677347543009\
 
>       258394021449002765441 - (2128273338395312794082239889502042426740163265\
 
                                                         7
>         896952213919734289338967670417476976749752274 x ) / 
 
>      295492331731770737179004968447431561595972282836347207013486677347543009\
 
>       258394021449002765441 + (3131531751677592541050434682573002978081935713\
 
                                      8
>         30118611843283471467300020 x ) / 
 
>      318301988696218072328823197473874700581978063256706855809825811905303940\
 
>       23, -(30561023295194185355473936080069440786472210733963618448825386011\
 
>         40982981087771271847186849 / 
 
>        2954923317317707371790049684474315615959722828363472070134866773475430\
 
>         09258394021449002765441) + 
 
>     (403820096205611425006263655037353780584002770792329800979997395647911001\
 
>         27798914167843551 x) / 
 
>      984974439105902457263349894824771871986574276121157356711622257825143364\
 
>       19464673816334255147 - (16224061591043681552944781085636984634391172763\
 
                                                        2
>         84519533340007849305542697483747684326137924 x ) / 
 
>      295492331731770737179004968447431561595972282836347207013486677347543009\
 
>       258394021449002765441 + (8918455245946849223188769574511865238577300197\
 
                                                        3
>         03980962783226696883987583952440973256029534 x ) / 
 
>      984974439105902457263349894824771871986574276121157356711622257825143364\
 
>       19464673816334255147 - (65801272978937180862712959407352590327040455244\
 
                                                       4
>         9234715337764352884110794834381405058719000 x ) / 
 
>      984974439105902457263349894824771871986574276121157356711622257825143364\
 
>       19464673816334255147 - (28901537351883956289136039912440745782245689529\
 
                                                       5
>         5252437382743008053813179629564451223353584 x ) / 
 
>      295492331731770737179004968447431561595972282836347207013486677347543009\
 
>       258394021449002765441 + (3502262857270097889869383807368973789349369548\
 
                                                         6
>         487952850057964117492722958613754374500895548 x ) / 
 
>      295492331731770737179004968447431561595972282836347207013486677347543009\
 
>       258394021449002765441 - (1954656349073869670376743415399869280016273993\
 
                                                        7
>         43132323550290087888300311598784623844035187 x ) / 
 
>      984974439105902457263349894824771871986574276121157356711622257825143364\
 
>       19464673816334255147 + (12344266025296865339187936601473975365262588163\
 
                                                       8
>         8734667380959858210897742268639106038815942 x ) / 
 
>      295492331731770737179004968447431561595972282836347207013486677347543009\
 
>       258394021449002765441 - (2192072226174314778735304277801102084657354999\
 
                                      9
>         31083028290298430027110014 x ) / 
 
>      318301988696218072328823197473874700581978063256706855809825811905303940\
 
>       23}

In[22]:= MakeMonic[f_, x_] returns the monic associate of f with respect to the variable
   x.

In[22]:=                       (* Using normalization of the remainders makes their coefficients
much smaller. Observe: *)\
ExtendedGeneric[f, g, qr2, Function[MakeMonic[#,x]]]
                       2    3      4      5      6       7    8      9      10
Dividend=-6 - 8 x - 6 x  + x  + 6 x  - 8 x  + 7 x  + 10 x  - x  - 4 x  - 7 x
 
                           2      3      4      5      6    7    8       9
>   , Divisor=7 + 7 x + 7 x  + 9 x  - 9 x  - 5 x  + 3 x  - x  + x  - 10 x , sy=0
                      2      3      4      5      6    7    8       9
Dividend=7 + 7 x + 7 x  + 9 x  - 9 x  - 5 x  + 3 x  - x  + x  - 10 x , Divisor=
 
                        2        3        4       5        6        7
    929   1619 x   129 x    813 x    393 x    65 x    909 x    837 x     8
>   --- + ------ + ------ + ------ - ------ - ----- - ------ - ------ + x , sy=
    77      77       7        77       77      77       77       77
 
      100
>   -(---)
      77
                             2        3        4       5        6        7
         929   1619 x   129 x    813 x    393 x    65 x    909 x    837 x     8
Dividend=--- + ------ + ------ + ------ - ------ - ----- - ------ - ------ + x
         77      77       7        77       77      77       77       77
 
                                            2          3          4         5
                77457    141832 x   130559 x    78882 x    26865 x    8713 x
>   , Divisor=-(-----) - -------- - --------- - -------- + -------- + ------- + 
                76471     76471       76471      76471      76471      76471
 
            6
     75706 x     7     8293    770 x
>    -------- + x , sy=----- + -----
      76471            76471   76471
                                       2          3          4         5
           77457    141832 x   130559 x    78882 x    26865 x    8713 x
Dividend=-(-----) - -------- - --------- - -------- + -------- + ------- + 
           76471     76471       76471      76471      76471      76471
 
            6                                                2            3
     75706 x     7            1970302    527825 x   1309484 x    1203236 x
>    -------- + x , Divisor=-(-------) - -------- - ---------- - ---------- - 
      76471                   6746007    2248669     6746007      6746007
 
            4            5                                        2
     62761 x    3945651 x     6      950387    833531 x   382355 x
>    -------- - ---------- + x , sy=-------- - -------- + ---------
      118351     4497338            13492014   13492014    6746007
                                          2            3          4
           1970302    527825 x   1309484 x    1203236 x    62761 x
Dividend=-(-------) - -------- - ---------- - ---------- - -------- - 
           6746007    2248669     6746007      6746007      118351
 
              5                                                        2
     3945651 x     6            370955408     892135202 x   880842616 x
>    ---------- + x , Divisor=-(----------) - ----------- - ------------ - 
      4497338                   1811113305    1811113305     1811113305
 
                3               4
     400194752 x    1206014702 x     5
>    ------------ + ------------- + x , sy=
      1811113305     1811113305
 
                                           2            3
       18321007     14544771 x   34959172 x    8994676 x
>   -(----------) + ---------- - ----------- - ----------
      1811113305    603704435    1811113305    362222661
                                                  2              3
           370955408     892135202 x   880842616 x    400194752 x
Dividend=-(----------) - ----------- - ------------ - ------------ + 
           1811113305    1811113305     1811113305     1811113305
 
                 4
     1206014702 x     5
>    ------------- + x , Divisor=
      1811113305
 
                                                   2                3
      443558207417    33896877604 x   47103283777 x    24075431738 x     4
>   -(------------) - ------------- - -------------- - -------------- + x , sy=
      523892406443     30817200379     74841772349      523892406443
 
                                               2                3
    39989946182    10563493593 x   2040647267 x    13871260325 x
>   ------------ - ------------- + ------------- - -------------- + 
    523892406443   523892406443    523892406443     523892406443
 
                  4
     18111133050 x
>    --------------
      523892406443
                                                        2                3
           443558207417    33896877604 x   47103283777 x    24075431738 x     4
Dividend=-(------------) - ------------- - -------------- - -------------- + x
           523892406443     30817200379     74841772349      523892406443
 
                                                                     2
              180886166026652   516943386113403 x   482636592397456 x     3
>   , Divisor=--------------- + ----------------- + ------------------ + x , sy=
              200546512066519    200546512066519     200546512066519
 
                                                          2
      29302481954429     17224372308151 x   869205094831 x
>   -(---------------) - ---------------- - --------------- - 
      200546512066519    200546512066519    200546512066519
 
                    3                 4                   5
     4491373072875 x    849414267189 x    15716772193290 x
>    ---------------- + --------------- - -----------------
     200546512066519    200546512066519    200546512066519
                                                                2
         180886166026652   516943386113403 x   482636592397456 x     3
Dividend=--------------- + ----------------- + ------------------ + x , Divisor=
         200546512066519    200546512066519     200546512066519
 
    34942067744154909   55274001465106507 x    2
>   ----------------- + ------------------- + x , sy=
    68972354977131950    34486177488565975
 
                                                                 2
      3608406498718972     216730018821043 x   1012748083042846 x
>   -(-----------------) - ----------------- + ------------------- - 
      34486177488565975    6897235497713195     34486177488565975
 
                      3                    4                     5
     986101381440742 x    344714118766977 x    5026912436041079 x
>    ------------------ + ------------------ - ------------------- + 
     34486177488565975    13794470995426390     68972354977131950
 
                      6
     200546512066519 x
>    ------------------
      6897235497713195
         34942067744154909   55274001465106507 x    2
Dividend=----------------- + ------------------- + x , Divisor=
         68972354977131950    34486177488565975
 
    5867929817485903777           735434497872964882     173972426284271020 x
>   ------------------- + x, sy=-(-------------------) + -------------------- + 
    9283395713049833367           9283395713049833367    3094465237683277789
 
                        2                       3                       4
     41309821422570476 x    341322421621392302 x    151135678718609255 x
>    -------------------- - --------------------- + --------------------- - 
     9283395713049833367     9283395713049833367     9283395713049833367
 
                         5                       6                       7
     177015684485767804 x    587226192139631045 x    344861774885659750 x
>    --------------------- + --------------------- - ---------------------
      3094465237683277789     9283395713049833367     9283395713049833367
         5867929817485903777
Dividend=------------------- + x, Divisor=1, sy=
         9283395713049833367
 
                                                                            2
    138616703212308147872   33864043328650605066 x   155804539965389705180 x
>   --------------------- + ---------------------- + ------------------------ - 
    534577883777518225711   534577883777518225711     534577883777518225711
 
                          3                          4
     1202108460219046540 x    229674853689311049421 x
>    ---------------------- - ------------------------ + 
      48597989434319838701     534577883777518225711
 
                            5                          6
     168110767023988104405 x    124342322713302765778 x
>    ------------------------ - ------------------------ + 
      534577883777518225711      534577883777518225711
 
                            7                          8
     135925387775817742274 x    185667914260996667340 x
>    ------------------------ - ------------------------
      534577883777518225711      534577883777518225711
3.58333 Second

             138616703212308147872   33864043328650605066 x
Out[22]= {1, --------------------- + ---------------------- + 
             534577883777518225711   534577883777518225711
 
                             2                        3
      155804539965389705180 x    1202108460219046540 x
>     ------------------------ - ---------------------- - 
       534577883777518225711      48597989434319838701
 
                             4                          5
      229674853689311049421 x    168110767023988104405 x
>     ------------------------ + ------------------------ - 
       534577883777518225711      534577883777518225711
 
                             6                          7
      124342322713302765778 x    135925387775817742274 x
>     ------------------------ + ------------------------ - 
       534577883777518225711      534577883777518225711
 
                             8
      185667914260996667340 x   195182586150195301849   703379446506471423 x
>     ------------------------, --------------------- - -------------------- + 
       534577883777518225711    534577883777518225711   48597989434319838701
 
                             2                          3
      103617417146431877924 x    170877179983932358602 x
>     ------------------------ - ------------------------ + 
       534577883777518225711      534577883777518225711
 
                             4                        5
      126074927281871157000 x    1678036625448472144 x
>     ------------------------ + ---------------------- - 
       534577883777518225711      48597989434319838701
 
                             6                         7
      223677301396921875548 x    37451123042774340561 x
>     ------------------------ + ----------------------- - 
       534577883777518225711      534577883777518225711
 
                           8                          9
      7883851740403985942 x    129967539982697667138 x
>     ---------------------- + ------------------------}
      534577883777518225711     534577883777518225711

In[23]:=