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Development of Algorithmic Constructions

08:43:05 16.Apr 2024

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Polynom = x^2+20x-3

1. Algorithm

a:=1;
b:=20;
c:=-3;


liste_max:=100000;

sieving:=proc (stelle, p)
begin
   while (stelle<=liste_max) do
     erg:=liste[stelle];
     while(erg mod p=0) do
        // Divison of the stored f(x) by the prime
        erg:=erg /p;
     end_while;
     liste[stelle]:=erg;
     stelle:=stelle+p;
   end_while;
end_proc;

// Calculation of the values of the polynom for x from 0 to liste_max
for x from 0 to liste_max do
   p:=abs (a*x^2+b*x+c);
   while (p mod 2=0) p:=p/2;
   liste [x]:=p;
end_for;

for x from 0 to liste_max do
    p:=liste[x];

    if (p>1) then

      // Printing the Primes
      print (x, p);

      // 1. Sieving
      sieving (x+p,  p);

      t:=(-x-b/a) mod p;
      if t=0 then t:=p; end_if;

      // 2. Sieving
      sieving (t, p);

    end_if;

end_for;

 2. Mathematical background
Lemma: If p | f(x) then also p | f(x+p) and  p | f(-x-b/a)

a)  p | f(x)   <=> ax^2 + bx + c = 0                       mod p
    p | f(x+p) <=> a(x+p)^2 + b(x+p) + c = 0               mod p
               <=> ax^2 + 2axp + ap^2 + bx + bp + c = 0    mod p
               <=> ax^2 + bx + c = 0                       mod p

    Thus if p | f(x) then p | f(x+p)

b)  if b = 0 mod a
    p | f(x)       <=> ax^2 + bx + c = 0                         mod p
    p | f(-x-b/a)  <=> a(-x-b/a)^2 + b(-x-b/a) + c = 0           mod p
                   <=> ax^2 + 2bx + b^2/a - bx - b^2/a + c = 0   mod p
                   <=> ax^2 + bx + c = 0                         mod p

    Thus if p | f(x)  then p | f(-x-b/a)



 3. Correctness of the algorithm
The proof for this polynom is similar to the proof for the polynom f(x)=x^2-4x+1.

a) First terms for the polynom f(x) = x^2+20x-3

f(0)=3
f(1)=1
f(2)=41
f(3)=11
f(4)=31
f(5)=61
f(6)=17
f(7)=1
f(8)=13
f(9)=43
f(10)=1
f(11)=1
f(12)=127
f(13)=71
f(14)=1
f(15)=29
f(16)=191
f(17)=313
f(18)=227
f(19)=1
f(20)=797
f(21)=1
f(22)=307
f(23)=1
f(24)=1
f(25)=1
f(26)=1193
f(27)=211
f(28)=149
f(29)=709
f(30)=499
f(31)=263
f(32)=151
f(33)=97
f(34)=47
f(35)=1
f(36)=1
f(37)=1
f(38)=1
f(39)=383
f(40)=1
f(41)=1249
f(42)=1
f(43)=1
f(44)=1
f(45)=487
f(46)=337
f(47)=1
f(48)=1087
f(49)=563
f(50)=269
f(51)=67
f(52)=1
f(53)=1933
f(54)=1
f(55)=229
f(56)=4253
f(57)=1
f(58)=137
f(59)=1
f(60)=1
f(61)=823
f(62)=5081
f(63)=1
f(64)=199
f(65)=251
f(66)=1
f(67)=971
f(68)=5981
f(69)=1
f(70)=2099
f(71)=3229
f(72)=2207
f(73)=1
f(74)=409
f(75)=1187
f(76)=1
f(77)=3733
f(78)=283
f(79)=1303
f(80)=727
f(81)=1
f(82)=929
f(83)=4273
f(84)=1
f(85)=1487
f(86)=701
f(87)=1
f(88)=3167
f(89)=373
f(90)=3299
f(91)=1
f(92)=10301
f(93)=103
f(94)=3571
f(95)=1
f(96)=1237
f(97)=1
f(98)=1051
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+20x-3  could be written as f(y)= y^2-103 with x=y-10

c) Backsubstitution

Beside by backsubstitution you get an estimation for the huge of the primes with p | f(x) and p < f(x)

f'(y)>(2y-1) with with y=x+10
f'(x)>2x+19

4. Infinity of the sequence

5. Sequence of the polynom with 1

6. Sequence of the polynom (only primes)

7. Distribution of the primes

8. Check for existing Integer Sequences by OEIS

Found in Database : 3, 1, 41, 11, 31, 61, 17, 1, 13, 43, 1, 1, 127, 71, 1, 29, 191, 313, 227, 1,
Found in Database : 3, 41, 11, 31, 61, 17, 13, 43, 127, 71, 29, 191, 313, 227, 797, 307, 1193, 211, 149, 709, 499, 263, 151, 97, 47, 383,
Found in Database : 3, 11, 13, 17, 29, 31, 41, 43, 47, 61, 67, 71, 97, 103, 127, 137, 149,