Search for polynominals
Inhaltsverzeichnis

Development of
Algorithmic Constructions

05:13:30
DeutschEnglish
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.-1634nē+2n+41
2.-674nē+2n+17
3.-434nē+2n+11
4.-194nē+2n+5
5.-114nē+2n+3
6.-74nē+7
7.-34nē+3
8.-24nē+4n+3
9.24nē+4n-1
10.34nē-3
11.54nē-5
12.74nē-7
13.114nē-11
14.134nē+2n-3
15.174nē-17
16.194nē+16n-3
17.234nē-23
18.294nē+2n-7
19.314nē-24n+5
20.374nē+6n-7
21.414nē-24n-5
22.434nē+52n-3
23.474nē-47
24.534nē+2n-13
25.594nē-32n+5
26.614nē+6n-13
27.674nē+24n-31
28.714nē-32n-7
29.734nē-68n-3
30.834nē-83
31.894nē+40n+11
32.974nē-40n+3
33.1014nē+6n-23
34.1034nē-40n-3
35.1074nē-40n-7
36.1094nē+22n+3
37.1494nē+2n-37
38.1574nē+18n-19
39.1674nē-167
40.1734nē+2n-43
41.1974nē+6n-47
42.2274nē-24n-191
43.2694nē-30n-11
44.2774nē-30n-13
45.2934nē+2n-73
46.3074nē-72n+17
47.3174nē+2n-79
48.3494nē+30n-31
49.3734nē-38n-3
50.3834nē-80n+17
51.4614nē-42n-5
52.4794nē-88n+5
53.5034nē-88n-19
54.5414nē-46n-3
55.5574nē-46n-7
56.6774nē+6n-167
57.7734nē-54n-11
58.7874nē-112n-3
59.8214nē-54n-23
60.8294nē-58n+3
61.8534nē-58n-3
62.9414nē-62n+5
63.10694nē+66n+5
64.14934nē-78n+7
65.16374nē-82n+11
66.16934nē-82n-3
67.18774nē-86n-7
68.23514nē-192n-47
69.26214nē-102n-5
70.30374nē-110n-3
71.42534nē-130n-7
72.72134nē-170n+3
73.94374nē-194n-7
74.141734nē-238n-3
75.723134nē-269n+3