Inhaltsverzeichnis

Development of
Algorithmic Constructions

11:20:11
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28.Mar 2024

Primes on quadric irreducible polynomials

The polynomials which are listed should fulfill the following conditions:
  1. The quadratic polynomials should be of the form f(x)=ax^2+bx+c, where a, b and c are elements of Z.
  2. The sequences start with x=0 and |f(0)|=|c| should be a prime number or equal 1.
  3. The polynomial should be irreducible, that means it should not be written as f(x)=(x+n)*(x+m), where n and m are integers.
  4. Every polynomial constructs a special sequence of primes which is infinite.
  5. The primes of the sequence are sorted in the list below.
  6. There are 178 different sequences.

Sort order

  1. I started the search for the polynoms from a=1 to a=4.
    For each value of a i made a loop of c=1, -1, 2, -2, 3, -3, 5, -5 up to 100, -100.
    For each value of a and b i made a loop for b from b=0, 1, -1, 2, -2, 3, -3, up to 300, -300.
    I only choose sequences with b=0 mod a because of the special mathematical proof.
    The first appearance of a sequence was choosen, all sequences were sorted and by that proceeding the table was created.
  2. |c| indicates the category. The primes p with p=f(x) are marked with green color.
  3. The prime sequences are sorted by arising primes.
You will find a better explanation of the algorithm, a mathematical proof and the distribution of the primes by following the link of the polynomial.

Results

Sequences
started with a=1
f(0)=239
f(0)=314
f(0)=58
f(0)=76
f(0)=113
f(0)=132
f(0)=17
f(0)=191
f(0)=23
f(0)=29
All73
Sequences
started with a=2
f(0)=2
f(0)=329
f(0)=513
f(0)=74
f(0)=116
f(0)=132
f(0)=171
f(0)=191
f(0)=23
f(0)=291
All57
Sequences
started with a=3
f(0)=226
f(0)=3
f(0)=54
f(0)=75
f(0)=112
f(0)=132
f(0)=171
f(0)=19
f(0)=23
f(0)=29
All40
Sequences
started with a=4
f(0)=2
f(0)=35
f(0)=51
f(0)=71
f(0)=11
f(0)=131
f(0)=17
f(0)=19
f(0)=23
f(0)=29
All8

Irreducible quadratic polynomials of the form ax^2+bx+c

x^2+bx+c Sequence start with 2

PolynomPolynomb^2-4acPrimedensity2357111317192329313741434753596167717379838997
1./1./1. f(x)=x^2+4x-2f(x+2)=x^2-6242 3 5 19 23 29 43 47 53 67 71 73 97
2./2./2. f(x)=x^2-6x+2f(x-3)=x^2-7282 3 7 19 29 31 37 47 53 59 83
3./3./3. f(x)=x^2-2x-2f(x-1)=x^2-3122 3 11 13 23 37 47 59 61 71 73 83 97
4./4./4. f(x)=x^2+2f(x+0)=x^2+2-82 3 11 17 19 41 43 59 67 73 83 89 97
5./5./5. f(x)=x^2-8x+2f(x-4)=x^2-14562 5 7 11 13 31 43 47 61 67
6./6./6. f(x)=x^2+6x-2f(x+3)=x^2-11442 5 7 11 19 37 43 53 79 83 89 97
7./7./7. f(x)=x^2-2x+2f(x-1)=x^2+1-42 5 13 17 29 37 41 53 61 73 89 97
8./8./8. f(x)=x^2-10x+2f(x-5)=x^2-23922 7 11 13 19 23 29 41 43 67 73 79 83
9./9./9. f(x)=x^2+4x+2f(x+2)=x^2-282 7 17 23 31 41 47 71 73 79 89 97
10./10./10. f(x)=x^2+12x-2f(x+6)=x^2-38152 0.700292 11 13 17 19 29 31 37 43 53 71 73 79 83
11./11./11. f(x)=x^2-14x+2f(x-7)=x^2-47188 0.701962 11 17 19 23 31 37 43 47 53 61 67 89 97
12./12./12. f(x)=x^2-16x+2f(x-8)=x^2-622482 13 19 23 29 31 37 41 53 59 61 67 79 97
13./13./13. f(x)=x^2+18x-2f(x+9)=x^2-83332 0.710142 17 19 29 37 41 43 47 61 67 71 79 83
14./14./14. f(x)=x^2-26x+2f(x-13)=x^2-167668 0.725402 23 29 43 59 61 67 71 79 83 89 97
15./15./15. f(x)=x^2+30x-2f(x+15)=x^2-227908 0.734902 29 31 53 67 73 83 89 97
16./16./16. f(x)=x^2-40x+2f(x-20)=x^2-3981592 0.757132 37 43 71 89

2x^2+bx+c Sequence start with 3

PolynomPolynomb^2-4acPrimedensity2357111317192329313741434753596167717379838997
1./1./17. f(x)=2x^2+16x-3f(x+4)=2*x^2-35280 0.702963 5 7 11 17 23 31 37 53 61 73 83 97
2./2./18. f(x)=2x^2-28x+3f(x-7)=2*x^2-95760 0.696323 5 7 11 19 23 29 31 47 67 71 79 97
3./3./19. f(x)=2x^2+3f(x+0)=2*x^2+3-24 0.707293 5 7 11 29 31 53 59 73 79 83 97
4./4./20. f(x)=2x^2-12x+3f(x-3)=2*x^2-15120 0.713093 5 7 13 17 19 29 37 71 83
5./5./21. f(x)=2x^2+64x-3f(x+16)=2*x^2-5154120 0.690903 5 7 13 19 23 31 41 43 59 67 71 73
6./6./22. f(x)=2x^2-20x-3f(x-5)=2*x^2-53424 0.696743 5 7 17 19 47 53 61 67 83 89 97
7./7./23. f(x)=2x^2-50x+3f(x-12.5)=2*x^2-309.52476 0.694183 5 11 19 37 41 43 47 53 59 61 67 73 83 89
8./8./24. f(x)=2x^2-20x+3f(x-5)=2*x^2-47376 0.700153 5 13 17 23 29 31 47 59 83 89 97
9./9./25. f(x)=2x^2-70x-3f(x-17.5)=2*x^2-615.54924 0.694613 5 13 23 29 37 41 47 59 67 71 79 83
10./10./26. f(x)=2x^2+8x+3f(x+2)=2*x^2-540 0.716553 5 13 31 37 41 43 53 67 71 79 83 89
11./11./27. f(x)=2x^2-52x+3f(x-13)=2*x^2-3352680 0.706033 5 19 23 31 37 43 47 59 61 67 79 89 97
12./12./28. f(x)=2x^2+12x-3f(x+3)=2*x^2-21168 0.715303 7 11 13 17 19 29 41 47 53 61 79 89
13./13./29. f(x)=2x^2+16x+3f(x+4)=2*x^2-29232 0.705013 7 11 19 23 29 37 43 61 71
14./14./30. f(x)=2x^2-62x-3f(x-15.5)=2*x^2-483.53868 0.699563 7 17 19 23 43 47 53 61 67 73 79 97
15./15./31. f(x)=2x^2-68x-3f(x-17)=2*x^2-5814648 0.716253 7 23 43 47 53 59 61 67 73 83 89 97
16./16./32. f(x)=2x^2+52x-3f(x+13)=2*x^2-3412728 0.709983 11 17 19 31 37 47 53 71 73 79 97
17./17./33. f(x)=2x^2+28x-3f(x+7)=2*x^2-101808 0.706483 11 17 23 29 31 47 53 59 61 67 71 79 83 97
18./18./34. f(x)=2x^2-106x+3f(x-26.5)=2*x^2-1401.511212 0.715343 11 17 37 53 61 67 79 83 89
19./19./35. f(x)=2x^2-104x+3f(x-26)=2*x^2-134910792 0.712333 11 19 29 31 37 41 61 71 73 97
20./20./36. f(x)=2x^2+32x-3f(x+8)=2*x^2-1311048 0.707893 11 23 29 31 37 41 43 47 59 71 79 89
21./21./37. f(x)=2x^2-64x+3f(x-16)=2*x^2-5094072 0.706703 13 17 19 23 53 59 61 71 73 89 97
22./22./38. f(x)=2x^2+46x+3f(x+11.5)=2*x^2-261.52092 0.709763 13 17 29 41 47 53 59 67 73 79 83 89
23./23./39. f(x)=2x^2+112x-3f(x+28)=2*x^2-157112568 0.719113 13 19 37 53 59 67 71 73
24./24./40. f(x)=2x^2-24x+3f(x-6)=2*x^2-69552 0.724283 17 19 23 29 31 37 43 59 61 67 73 89
25./25./41. f(x)=2x^2-158x-3f(x-39.5)=2*x^2-3123.524988 0.740243 17 31 53 61 67 89
26./26./42. f(x)=2x^2+172x-3f(x+43)=2*x^2-370129608 0.740773 19 29 43 59 71 79
27./27./43. f(x)=2x^2-76x+3f(x-19)=2*x^2-7195752 0.731903 23 47 53 59 71 79 83
28./28./44. f(x)=2x^2-36x+3f(x-9)=2*x^2-1591272 0.740623 29 31 41 43 47 53 61 79 83 97
29./29./45. f(x)=2x^2+134x+3f(x+33.5)=2*x^2-2241.517932 0.753653 31 43 47 71 83

2x^2+bx+c Sequence start with 5

PolynomPolynomb^2-4acPrimedensity2357111317192329313741434753596167717379838997
1./30./46. f(x)=2x^2+5f(x+0)=2*x^2+5-40 0.711245 7 11 13 19 23 37 41 47 53 59 89
2./31./47. f(x)=2x^2-24x+5f(x-6)=2*x^2-67536 0.707985 7 13 17 19 31 53 59 61 67 73 79 83 89
3./32./48. f(x)=2x^2+16x-5f(x+4)=2*x^2-37296 0.711085 7 13 19 29 37 41 43 47 59 61 71 73
4./33./49. f(x)=2x^2-32x-5f(x-8)=2*x^2-1331064 0.715505 7 19 23 29 31 37 41 53 61 67 83 89 97
5./34./50. f(x)=2x^2-42x+5f(x-10.5)=2*x^2-215.51724 0.712865 7 29 31 41 43 47 53 61 67 71 79 83 97
6./35./51. f(x)=2x^2+12x+5f(x+3)=2*x^2-13104 0.717845 11 13 17 19 23 37 59 67 79 83
7./36./52. f(x)=2x^2-54x+5f(x-13.5)=2*x^2-359.52876 0.706895 11 13 19 23 29 37 43 47 61 67 71 79
8./37./53. f(x)=2x^2+20x-5f(x+5)=2*x^2-55440 0.724665 11 17 23 29 37 43 47 53 59 61 73 79 83 89
9./38./54. f(x)=2x^2+64x-5f(x+16)=2*x^2-5174136 0.735705 11 41 47 61 67 71 73 83 89 97
10./39./55. f(x)=2x^2+56x-5f(x+14)=2*x^2-3973176 0.721545 13 23 31 47 53 59 61 73 79 97
11./40./56. f(x)=2x^2+72x+5f(x+18)=2*x^2-6435144 0.731165 13 37 47 61 71 79 83 89 97
12./41./57. f(x)=2x^2-28x-5f(x-7)=2*x^2-103824 0.719025 17 19 31 37 41 47 53 59 71 83 97
13./42./58. f(x)=2x^2+36x+5f(x+9)=2*x^2-1571256 0.722195 17 29 31 43 47 53 59 61 71 83 89

2x^2+bx+c Sequence start with 7

PolynomPolynomb^2-4acPrimedensity2357111317192329313741434753596167717379838997
1./43./59. f(x)=2x^2-24x-7f(x-6)=2*x^2-79632 0.720417 11 19 29 37 47 53 61 67 71 73 79 83 89 97
2./44./60. f(x)=2x^2-28x+7f(x-7)=2*x^2-91728 0.733427 13 19 37 41 43 59 61 71 73 83 89 97
3./45./61. f(x)=2x^2-54x-7f(x-13.5)=2*x^2-371.52972 0.746477 37 41 53 59 67 79 97
4./46./62. f(x)=2x^2+58x+7f(x+14.5)=2*x^2-413.53308 0.750087 43 47 59 61 67 73

2x^2+bx+c Sequence start with 11

PolynomPolynomb^2-4acPrimedensity2357111317192329313741434753596167717379838997
1./47./63. f(x)=2x^2-52x-11f(x-13)=2*x^2-3492792 0.7226011 13 17 23 31 41 43 53 59 61 73
2./48./64. f(x)=2x^2+11f(x+0)=2*x^2+11-88 0.7243911 13 19 23 29 31 43 47 61 71 83 89 97
3./49./65. f(x)=2x^2-38x-11f(x-9.5)=2*x^2-191.51532 0.7395111 17 29 47 59 73 79 83
4./50./66. f(x)=2x^2-92x-11f(x-23)=2*x^2-10698552 0.7563311 17 43 61 71 79 83 89 97
5./51./67. f(x)=2x^2+24x+11f(x+6)=2*x^2-61488 0.7354211 29 37 41 43 47 53 59 61 67 73 97
6./52./68. f(x)=2x^2-62x-11f(x-15.5)=2*x^2-491.53932 0.7623111 37 41 53 71 83 89

2x^2+bx+c Sequence start with 13

PolynomPolynomb^2-4acPrimedensity2357111317192329313741434753596167717379838997
1./53./69. f(x)=2x^2+44x+13f(x+11)=2*x^2-2291832 0.7446313 17 29 59 67 71 97
2./54./70. f(x)=2x^2-46x+13f(x-11.5)=2*x^2-251.52012 0.7479113 19 31 61 71 73 97

2x^2+bx+c Sequence start with 17

PolynomPolynomb^2-4acPrimedensity2357111317192329313741434753596167717379838997
1./55./71. f(x)=2x^2-78x+17f(x-19.5)=2*x^2-743.55948 0.7597717 19 37 43 59 73 79 97

2x^2+bx+c Sequence start with 19

PolynomPolynomb^2-4acPrimedensity2357111317192329313741434753596167717379838997
1./56./72. f(x)=2x^2+40x+19f(x+10)=2*x^2-1811448 0.7666019 53 61 73 79 83

2x^2+bx+c Sequence start with 29

PolynomPolynomb^2-4acPrimedensity2357111317192329313741434753596167717379838997
1./57./73. f(x)=2x^2+29f(x+0)=2*x^2+29-232 0.7581329 31 37 47 59 61 67 79 83

3x^2+bx+c Sequence start with 2

PolynomPolynomb^2-4acPrimedensity2357111317192329313741434753596167717379838997
1./1./74. f(x)=3x^2-6x-2f(x-1)=3*x^2-5602 5 7 11 17 43 53 59 61 67 71
2./2./75. f(x)=3x^2+2f(x+0)=3*x^2+2-24 0.717522 5 7 11 29 31 53 59 73 79 83 97
3./3./76. f(x)=3x^2-12x+2f(x-2)=3*x^2-10120 0.723272 5 7 13 17 19 29 37 71 83
4./4./77. f(x)=3x^2-2f(x+0)=3*x^2-2242 5 19 23 29 43 47 53 67 71 73 97
5./5./78. f(x)=3x^2-12x-2f(x-2)=3*x^2-14168 0.725442 7 11 13 17 19 29 41 47 53 61 79 89
6./6./79. f(x)=3x^2-6x+2f(x-1)=3*x^2-1122 11 13 23 37 47 59 61 71 73 83 97
7./7./80. f(x)=3x^2-18x-2f(x-3)=3*x^2-29348 0.729592 13 17 19 23 29 31 41 43 59 71 79 83 89
8./8./81. f(x)=3x^2-24x+2f(x-4)=3*x^2-46552 0.734342 17 19 23 29 31 37 43 59 61 67 73 89
9./9./82. f(x)=3x^2-36x+2f(x-6)=3*x^2-1061272 0.750522 29 31 41 43 47 53 61 79 83 97
10./10./83. f(x)=3x^2-42x-2f(x-7)=3*x^2-1491788 0.760372 37 41 43 47 61 73 79 89

4x^2+bx+c Sequence start with 3

PolynomPolynomb^2-4acPrimedensity2357111317192329313741434753596167717379838997
1./1./84. f(x)=4x^2+32x+3f(x+4)=4*x^2-619763 5 13 19 41 47 61 73 83 97
2./2./85. f(x)=4x^2-124x+3f(x-15.5)=4*x^2-95815328 0.721913 11 13 29 31 37 47 53 73 79 89 97
3./3./86. f(x)=4x^2-24x+3f(x-3)=4*x^2-33528 0.755403 11 17 29 31 37 41 67 83 97
4./4./87. f(x)=4x^2-40x+3f(x-5)=4*x^2-971552 0.742713 11 31 43 47 53 61 73 79 89 97
5./5./88. f(x)=4x^2+16x+3f(x+2)=4*x^2-132083 13 17 23 29 43 53 61 79

4x^2+bx+c Sequence start with 5

PolynomPolynomb^2-4acPrimedensity2357111317192329313741434753596167717379838997
1./6./89. f(x)=4x^2-24x-5f(x-3)=4*x^2-416565 23 31 37 41 43 59 61 73 83

4x^2+bx+c Sequence start with 7

PolynomPolynomb^2-4acPrimedensity2357111317192329313741434753596167717379838997
1./7./90. f(x)=4x^2+7f(x+0)=4*x^2+7-1127 11 23 29 37 43 53 67 71 79

4x^2+bx+c Sequence start with 13

PolynomPolynomb^2-4acPrimedensity2357111317192329313741434753596167717379838997
1./8./91. f(x)=4x^2-8x-13f(x-1)=4*x^2-1727213 17 19 43 47 53 59 67 83 89