Bribing a computer to compute the roots of (A4f) polynomial.

  Our (A4f) polynomial was

   h(x)  =  [x + 5] * [x - 11] * [x + 37].

In class, we saw its reduction mod 8, mod 3, and  mod 5,
and we found the respective roots.
 
  Promising my computer an extra mouse, I asked its help:

=========================================================

& (defun hof (x) (* (+ x 5) (- x 11) (+ x 37)))

& (setq CRT-8-3-5 (cree-crt 8 3 5))  =>

#S(CHINESE-FUSION::CRTSTR :EUCLDOM #1=<<RING: InTeGeRs>>
   :PROD 120 :MODULI #(8 3 5)
   :REDUCEDPRODS #(15 40 24) :MAGICNUMS #(105 40 96))

& (setq U (CRTSTR-PROD CRT-8-3-5)) => 120

& (setq roots8 '(1 3 5 7)   roots3 '(1 2)   roots5 '(0 1 3))

& (print-h-table roots8 roots3 roots5)


root8 root3 root5 | URoot |  h(URoot)  | Mod-U
    1     1     0 |    25 |      26040 |   0
    1     1     1 |     1 |      -2280 |   0
    1     1     3 |   -47 |     -24360 |   0
    1     2     0 |   -55 |     -59400 |   0
    1     2     1 |    41 |     107640 |   0
    1     2     3 |    -7 |       1080 |   0
    3     1     0 |    -5 |          0 |   0
    3     1     1 |   -29 |       7680 |   0
    3     1     3 |    43 |     122880 |   0
    3     2     0 |    35 |      69120 |   0
    3     2     1 |    11 |          0 |   0
    3     2     3 |   -37 |          0 |   0
    5     1     0 |   -35 |       2760 |   0
    5     1     1 |   -59 |     -83160 |   0
    5     1     3 |    13 |       1800 |   0
    5     2     0 |     5 |      -2520 |   0
    5     2     1 |   -19 |       7560 |   0
    5     2     3 |    53 |     219240 |   0
    7     1     0 |    55 |     242880 |   0
    7     1     1 |    31 |      48960 |   0
    7     1     3 |   -17 |       6720 |   0
    7     2     0 |   -25 |       8640 |   0
    7     2     1 |   -49 |     -31680 |   0
    7     2     3 |    23 |      20160 |   0

================

Recall: For modulus-tuple (8,3,5), the magic tuple is
:MAGICNUMS #(105 40 96))


Here is computation "by hand":

% (setq M1 8  M2 3  M3 5  U (* M1 M2 M3))	=>  120

% (setq R1 (/ U M1)  R2 (/ U M2)  R3 (/ U M3))


Here are the reduced products:

% (list R1 R2 R3)	=>	(15 40 24)



% (lightning R1 M1)

  n:   r_n   q_n   s_n   t_n
/---------------------------\
  0:    15    --     1     0
  1:     8     1     0     1
  2:     7     1     1    -1
  3:     1     7    -1     2
  4:     0 Infty     8   -15
\___________________________/
            1  =  [15]*[-1] + [8]*[2].

% (setq RecipR1 -1 )



% (lightning R2 M2)

  n:   r_n   q_n   s_n   t_n
/---------------------------\
  0:    40    --     1     0
  1:     3    13     0     1
  2:     1     3     1   -13
  3:     0 Infty    -3    40
\___________________________/
            1  =  [40]*[1] + [3]*[-13].

% (setq RecipR2 1 )


% (lightning R3 M3)

  n:   r_n   q_n   s_n   t_n
/---------------------------\
  0:    24    --     1     0
  1:     5     4     0     1
  2:     4     1     1    -4
  3:     1     4    -1     5
  4:     0 Infty     5   -24
\___________________________/
            1  =  [24]*[-1] + [5]*[5].

% (setq RecipR3 -1 )


Now we compute the magic numbers...

% (setq MagicList (list (* R1 RecipR1 ) (* R2 RecipR2 ) (* R3 RecipR3 )))	=>		(-15 40 576)


% (iter (for G in MagicList) (for j from 1) (format t "~%G_~D   =  ~3D" j (mod G U)))

  G_1   =  105
  G_2   =   40
  G_3   =   96


==== End of "chinese-rt3.easier.POLY.txt" ====