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Development of Algorithmic Constructions

17:20:00 18.Apr 2024

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Polynom = x^2+12x-109

1. Algorithm

a:=1;
b:=12;
c:=-109;


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-109

f(0)=109
f(1)=3
f(2)=1
f(3)=1
f(4)=5
f(5)=1
f(6)=1
f(7)=1
f(8)=17
f(9)=1
f(10)=37
f(11)=1
f(12)=179
f(13)=1
f(14)=1
f(15)=1
f(16)=113
f(17)=1
f(18)=431
f(19)=1
f(20)=59
f(21)=73
f(22)=71
f(23)=29
f(24)=151
f(25)=1
f(26)=293
f(27)=1
f(28)=337
f(29)=1
f(30)=1151
f(31)=1
f(32)=433
f(33)=43
f(34)=97
f(35)=1
f(36)=1619
f(37)=1
f(38)=199
f(39)=47
f(40)=1
f(41)=1
f(42)=127
f(43)=1
f(44)=157
f(45)=307
f(46)=853
f(47)=1
f(48)=163
f(49)=1
f(50)=997
f(51)=1
f(52)=1
f(53)=139
f(54)=691
f(55)=149
f(56)=137
f(57)=239
f(58)=439
f(59)=1
f(60)=4211
f(61)=181
f(62)=1493
f(63)=577
f(64)=317
f(65)=1
f(66)=5039
f(67)=1
f(68)=1777
f(69)=1
f(70)=1877
f(71)=241
f(72)=5939
f(73)=1
f(74)=1
f(75)=401
f(76)=1
f(77)=281
f(78)=6911
f(79)=1
f(80)=2417
f(81)=1
f(82)=1
f(83)=1
f(84)=1
f(85)=1
f(86)=1
f(87)=1063
f(88)=2897
f(89)=1
f(90)=193
f(91)=1
f(92)=1051
f(93)=1
f(94)=1
f(95)=419
f(96)=10259
f(97)=1
f(98)=3557
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+12x-109  could be written as f(y)= y^2-145 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 : 109, 3, 1, 1, 5, 1, 1, 1, 17, 1, 37, 1, 179, 1, 1, 1, 113, 1, 431, 1,
Found in Database : 109, 3, 5, 17, 37, 179, 113, 431, 59, 73, 71, 29, 151, 293, 337, 1151, 433, 43, 97, 1619, 199, 47,
Found in Database : 3, 5, 17, 29, 37, 43, 47, 59, 71, 73, 97, 109, 113, 127, 137, 139, 149,