1.Introduction

  At fourth polynomials f(x) =ax^4+bx^3+cx^2+dx+e, looking for those producing only primes
 in the x range [0, k]. 


                 T: Total primes(The quantity that absolute of f(x) is prime)
                 N: Negative primes in the total primes
                 D: Distinct primes in the total primes

  Search contents:

                 1. Find the maximum number of T.
                 2. Find the maximum number of D.
                 3. Find the maximum number of T(only positive primes).
 
  Next,I show the result which I searched.


2.Search results
      
     
                  No   a     b      c       d      e   T   N   D

                  1.   1  -134   6272 -119461 793669  68   4  34
                  2.   1   -63   1155   -7449  11593  43  31  42
                  3.   1   -63   1165   -7649  15073  42  25  42
                  4.   1   -70   1429   -7140  11287  38   0  20
                  5.   1   -74   1525   -5772  10159  38   0  19

                       1   -99   3441  -49259 247889  50   9  49    (20006.4.2)
                       1   -78   1873  -13728  31859  40   0  20    (20006.4.2) 

            
                (1). About No.1 polynomial: x^4-134*x^3+6272*x^2-119461*x+793669

                These polynomials are linked with each other.

　　            Let  f(x)=  x^4-74*x^3+1592*x^2-8251*x+11329

                   f(x-1)=  x^4-78*x^3+1820*x^2-11661*x+21247

                   f(x-2)=  x^4-82*x^3+2060*x^2-15539*x+34807

                   f(x-3)=  x^4-86*x^3+2312*x^2-19909*x+52489
                   f(x-4)=  x^4-90*x^3+2576*x^2-24795*x+74797
                   f(x-5)=  x^4-94*x^3+2852*x^2-30221*x+102259
                   f(x-6)=  x^4-98*x^3+3140*x^2-36211*x+135427
                   f(x-7)=  x^4-102*x^3+3440*x^2-42789*x+174877
                   f(x-8)=  x^4-106*x^3+3752*x^2-49979*x+221209
                   f(x-9)=  x^4-110*x^3+4076*x^2-57805*x+275047
                   f(x-10)= x^4-114*x^3+4412*x^2-66291*x+337039
                   f(x-11)= x^4-118*x^3+4760*x^2-75461*x+407857
                   f(x-12)= x^4-122*x^3+5120*x^2-85339*x+488197
                   f(x-13)= x^4-126*x^3+5492*x^2-95949*x+578779
                   f(x-14)= x^4-130*x^3+5876*x^2-107315*x+680347
                   f(x-15)= x^4-134*x^3+6272*x^2-119461*x+793669

   
       
3.Detail

                     1.  x^4-134*x^3+6272*x^2-119461*x+793669
                            T=68,N=4,D=34

                             x     f(x)

                             0   793669   
                             1   680347
                             2   578779
                             3   488197
                             4   407857
                             5   337039
                             6   275047
                             7   221209
                             8   174877
                             9   135427
                            10   102259
                            11    74797
                            12    52489
                            13    34807
                            14    21247
                            15    11329
                            16     4597
                            17      619
                            18    -1013
                            19     -683
                            20     1249
                            21     4447
                            22     8599
                            23    13417
                            24    18637
                            25    24019
                            26    29347
                            27    34429
                            28    39097
                            29    43207
                            30    46639
                            31    49297
                            32    51109
                            33    52027
                            34    52027
                            35    51109
                            36    49297
                            37    46639
                            38    43207
                            39    39097
                            40    34429
                            41    29347
                            42    24019
                            43    18637
                            44    13417
                            45     8599
                            46     4447
                            47     1249
                            48     -683
                            49    -1013
                            50      619
                            51     4597
                            52    11329
                            53    21247
                            54    34807
                            55    52489
                            56    74797
                            57   102259
                            58   135427
                            59   174877
                            60   221209
                            61   275047
                            62   337039
                            63   407857
                            64   488197
                            65   578779
                            66   680347
                            67   793669

            It is understood that it is symmetrical shape.



          f(x)=x^4-134*x^3+6272*x^2-119461*x+793669



                      2.  x^4-63*x^3+1155*x^2-7449*x+11593
                            T=43,N=31,D=42

                             x     f(x)
                             0    11593
                             1     5237
                             2      827
                             3    -1979
                             4    -3499
                             5    -4027
                             6    -3833
                             7    -3163
                             8    -2239
                             9    -1259
                            10     -397
                            11      197
                            12      397
                            13      101
                            14     -769
                            15    -2267
                            16    -4423
                            17    -7243
                            18   -10709
                            19   -14779
                            20   -19387
                            21   -24443
                            22   -29833
                            23   -35419
                            24   -41039
                            25   -46507
                            26   -51613
                            27   -56123
                            28   -59779
                            29   -62299
                            30   -63377
                            31   -62683
                            32   -59863
                            33   -54539
                            34   -46309
                            35   -34747
                            36   -19403
                            37      197
                            38    24551
                            39    54181
                            40    89633
                            41   131477
                            42   180307

 

                     4.  x^4-70*x^3+1429*x^2-7140*x+11287

                            T=38,N=0,D=20

                             x     f(x)
                             0    11287
                             1     5507
                             2     2179
                             3      919
                             4     1367
                             5     3187
                             6     6067
                             7     9719
                             8    13879
                             9    18307
                            10    22787
                            11    27127
                            12    31159
                            13    34739
                            14    37747
                            15    40087
                            16    41687
                            17    42499
                            18    42499
                            19    41687
                            20    40087
                            21    37747
                            22    34739
                            23    31159
                            24    27127
                            25    22787
                            26    18307
                            27    13879
                            28     9719
                            29     6067
                            30     3187
                            31     1367
                            32      919
                            33     2179
                            34     5507
                            35    11287
                            36    19927
                            37    31859