---

Development of Algorithmic Constructions

18:00:23 19.Apr 2024

---

Polynom = x^2+84x-109

1. Algorithm

a:=1;
b:=84;
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+84x-109

f(0)=109
f(1)=3
f(2)=7
f(3)=19
f(4)=1
f(5)=1
f(6)=431
f(7)=11
f(8)=1
f(9)=13
f(10)=277
f(11)=1
f(12)=149
f(13)=1
f(14)=421
f(15)=43
f(16)=71
f(17)=67
f(18)=157
f(19)=1
f(20)=73
f(21)=131
f(22)=1
f(23)=1
f(24)=191
f(25)=1
f(26)=1
f(27)=1
f(28)=1009
f(29)=1
f(30)=1
f(31)=1
f(32)=1201
f(33)=1
f(34)=1301
f(35)=1
f(36)=4211
f(37)=1
f(38)=503
f(39)=293
f(40)=1
f(41)=1
f(42)=1
f(43)=223
f(44)=263
f(45)=89
f(46)=103
f(47)=1
f(48)=479
f(49)=1
f(50)=1
f(51)=1
f(52)=211
f(53)=1
f(54)=1049
f(55)=1
f(56)=859
f(57)=991
f(58)=1
f(59)=347
f(60)=449
f(61)=1
f(62)=271
f(63)=1
f(64)=3121
f(65)=1
f(66)=9791
f(67)=139
f(68)=487
f(69)=653
f(70)=3557
f(71)=227
f(72)=1
f(73)=1
f(74)=1
f(75)=1
f(76)=1
f(77)=1
f(78)=12527
f(79)=1
f(80)=4337
f(81)=1657
f(82)=643
f(83)=1
f(84)=1
f(85)=1
f(86)=691
f(87)=1
f(88)=5009
f(89)=1
f(90)=15551
f(91)=659
f(92)=1787
f(93)=1
f(94)=1847
f(95)=1
f(96)=1
f(97)=727
f(98)=311
f(99)=2251

b) Substitution of the polynom
The polynom f(x)=x^2+84x-109  could be written as f(y)= y^2-1873 with x=y-42

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+42
f'(x)>2x+83

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, 7, 19, 1, 1, 431, 11, 1, 13, 277, 1, 149, 1, 421, 43, 71, 67, 157, 1,
Found in Database : 109, 3, 7, 19, 431, 11, 13, 277, 149, 421, 43, 71, 67, 157, 73, 131, 191, 1009, 1201, 1301, 4211, 503, 293,
Found in Database : 3, 7, 11, 13, 19, 43, 67, 71, 73, 89, 103, 109, 131, 139, 149,