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Development of Algorithmic Constructions

12:40:43 20.Apr 2024

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Polynom = x^2+12x-19

1. Algorithm

a:=1;
b:=12;
c:=-19;


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+12x-19

f(0)=19
f(1)=3
f(2)=1
f(3)=13
f(4)=5
f(5)=11
f(6)=89
f(7)=1
f(8)=47
f(9)=17
f(10)=67
f(11)=1
f(12)=269
f(13)=1
f(14)=23
f(15)=193
f(16)=1
f(17)=79
f(18)=521
f(19)=1
f(20)=1
f(21)=337
f(22)=1
f(23)=131
f(24)=1
f(25)=151
f(26)=1
f(27)=1
f(28)=367
f(29)=1
f(30)=73
f(31)=1
f(32)=463
f(33)=733
f(34)=103
f(35)=271
f(36)=1709
f(37)=1
f(38)=1
f(39)=197
f(40)=229
f(41)=359
f(42)=173
f(43)=1
f(44)=163
f(45)=1
f(46)=883
f(47)=1
f(48)=2861
f(49)=1
f(50)=1
f(51)=1597
f(52)=1103
f(53)=571
f(54)=709
f(55)=1
f(56)=421
f(57)=1
f(58)=449
f(59)=139
f(60)=1
f(61)=739
f(62)=1523
f(63)=181
f(64)=1
f(65)=277
f(66)=223
f(67)=293
f(68)=1
f(69)=557
f(70)=1907
f(71)=1
f(72)=6029
f(73)=1031
f(74)=1
f(75)=3253
f(76)=1
f(77)=1
f(78)=7001
f(79)=239
f(80)=2447
f(81)=1
f(82)=233
f(83)=1
f(84)=1609
f(85)=457
f(86)=2803
f(87)=4297
f(88)=2927
f(89)=1
f(90)=9161
f(91)=1559
f(92)=1061
f(93)=443
f(94)=1
f(95)=1
f(96)=1
f(97)=1759
f(98)=211
f(99)=1097

b) Substitution of the polynom
The polynom f(x)=x^2+12x-19  could be written as f(y)= y^2-55 with x=y-6

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+6
f'(x)>2x+11

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 : 19, 3, 1, 13, 5, 11, 89, 1, 47, 17, 67, 1, 269, 1, 23, 193, 1, 79, 521, 1,
Found in Database : 19, 3, 13, 5, 11, 89, 47, 17, 67, 269, 23, 193, 79, 521, 337, 131, 151, 367, 73, 463, 733, 103, 271, 1709, 197,
Found in Database : 3, 5, 11, 13, 17, 19, 23, 47, 67, 73, 79, 89, 103, 131, 139,