Search for polynominals

# Development ofAlgorithmic Constructions

 05:17:57 19.Sep 2021

## Prime generators on quadratic irreducible polynomials

This is a collection of different polynomials of the form f(n)=2n²+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. -37 2nē+2n+19 2. -13 2nē+2n+7 3. -5 2nē+2n+3 4. -2 2nē+1 5. 2 2nē-1 6. 3 2nē+2n-1 7. 7 2nē+2n-3 8. 11 2nē+2n-5 9. 19 2nē+6n-5 10. 23 2nē+2n-11 11. 31 2nē-10n-3 12. 43 2nē+14n+3 13. 47 2nē+2n-23 14. 59 2nē+10n-17 15. 71 2nē-18n+5 16. 83 2nē+2n-41 17. 107 2nē+10n-41 18. 127 2nē+18n-23 19. 131 2nē+22n-5 20. 163 2nē+26n+3 21. 167 2nē+2n-83 22. 193 2nē-13n-3 23. 211 2nē-30n+7 24. 227 2nē+2n-113 25. 239 2nē-30n-7 26. 263 2nē-34n+13 27. 283 2nē+30n-29 28. 347 2nē+22n-113 29. 367 2nē-38n-3 30. 383 2nē-38n-11 31. 409 2nē+15n-23 32. 419 2nē+38n-29 33. 431 2nē-42n+5 34. 479 2nē-42n-19 35. 503 2nē-46n+13 36. 523 2nē+42n-41 37. 563 2nē+34n-137 38. 619 2nē-50n+3 39. 719 2nē-54n+5 40. 743 2nē-54n-7 41. 827 2nē+58n+7 42. 887 2nē-58n-23 43. 937 2nē+31n+3 44. 947 2nē+54n-109 45. 967 2nē-62n-3 46. 983 2nē-62n-11 47. 1009 2nē+21n-71 48. 1163 2nē+62n-101 49. 1231 2nē-70n-3 50. 1483 2nē-78n+19 51. 1487 2nē-78n+17 52. 1609 2nē-39n-11 53. 1931 2nē+86n-41 54. 1993 2nē-39n-59 55. 2017 2nē+45n+1 56. 2377 2nē-49n+3 57. 2713 2nē-57n+67 58. 2803 2nē-106n+3 59. 3457 2nē+59n+3 60. 3793 2nē+51n-149 61. 4483 2nē+134n+3 62. 6089 2nē-81n+59 63. 6247 2nē-158n-3 64. 7417 2nē+81n-107 65. 14593 2nē+117n-113 66. 17209 2nē-135n+127 67. 28657 2nē+165n-179 68. 50377 2nē-225n+31 69. 56209 2nē-237n-5