Search for polynominals

# Development ofAlgorithmic Constructions

 05:13:30 19.Sep 2021

## Prime generators on quadratic irreducible polynomials

This is a collection of different polynomials of the form f(n)=4n²+bn+c .
The searched polynomials should fulfill the following conditions:
1. The polynomial should be irreducible.
2. The term abs(c) should be a prime which is a limitiation.
3. Every polynomial should construct a special infinite sequence of primes by the following described algorithm

## The algorithm

The following described algorithm is the simplest algorithm which calculates an infinite sequence of primes.
The algorithm can be improved of course by several means, but for a simple understanding the algorithm is limited to this simple form:
1. All positiv values f(n)=abs(n²+c) for n=0 up to n_max with n element N are precalculated.
2. All even values of f(n) are devided by 2 until they are odd.
3. for n=0 : f(0)=abs(c) and the prime abs (c) is sieved out for the values of f(0+k*(abs(c))) for k=1, 2, 3, 4, etc.
4. for n=1 : if f(n)=1 nothing is made if f(n)>1=p, p is a prime and is sieved out
for the values of f( n+k*p)) for k=1, 2, 3, 4, etc. and
for the values of f(-n+k*p)) for k=1, 2, 3, 4, etc.
5. n is increased by one and the last step is repeated until n=n_max.

## Results

Primzahlsystem
 1. -163 4nē+2n+41 2. -67 4nē+2n+17 3. -43 4nē+2n+11 4. -19 4nē+2n+5 5. -11 4nē+2n+3 6. -7 4nē+7 7. -3 4nē+3 8. -2 4nē+4n+3 9. 2 4nē+4n-1 10. 3 4nē-3 11. 5 4nē-5 12. 7 4nē-7 13. 11 4nē-11 14. 13 4nē+2n-3 15. 17 4nē-17 16. 19 4nē+16n-3 17. 23 4nē-23 18. 29 4nē+2n-7 19. 31 4nē-24n+5 20. 37 4nē+6n-7 21. 41 4nē-24n-5 22. 43 4nē+52n-3 23. 47 4nē-47 24. 53 4nē+2n-13 25. 59 4nē-32n+5 26. 61 4nē+6n-13 27. 67 4nē+24n-31 28. 71 4nē-32n-7 29. 73 4nē-68n-3 30. 83 4nē-83 31. 89 4nē+40n+11 32. 97 4nē-40n+3 33. 101 4nē+6n-23 34. 103 4nē-40n-3 35. 107 4nē-40n-7 36. 109 4nē+22n+3 37. 149 4nē+2n-37 38. 157 4nē+18n-19 39. 167 4nē-167 40. 173 4nē+2n-43 41. 197 4nē+6n-47 42. 227 4nē-24n-191 43. 269 4nē-30n-11 44. 277 4nē-30n-13 45. 293 4nē+2n-73 46. 307 4nē-72n+17 47. 317 4nē+2n-79 48. 349 4nē+30n-31 49. 373 4nē-38n-3 50. 383 4nē-80n+17 51. 461 4nē-42n-5 52. 479 4nē-88n+5 53. 503 4nē-88n-19 54. 541 4nē-46n-3 55. 557 4nē-46n-7 56. 677 4nē+6n-167 57. 773 4nē-54n-11 58. 787 4nē-112n-3 59. 821 4nē-54n-23 60. 829 4nē-58n+3 61. 853 4nē-58n-3 62. 941 4nē-62n+5 63. 1069 4nē+66n+5 64. 1493 4nē-78n+7 65. 1637 4nē-82n+11 66. 1693 4nē-82n-3 67. 1877 4nē-86n-7 68. 2351 4nē-192n-47 69. 2621 4nē-102n-5 70. 3037 4nē-110n-3 71. 4253 4nē-130n-7 72. 7213 4nē-170n+3 73. 9437 4nē-194n-7 74. 14173 4nē-238n-3 75. 72313 4nē-269n+3