---

Development of Algorithmic Constructions

15:28:16 18.Apr 2024

---

Polynom = x^2+43x-3

1. Algorithm

a:=1;
b:=43;
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+43x-3

f(0)=3
f(1)=41
f(2)=29
f(3)=5
f(4)=37
f(5)=79
f(6)=97
f(7)=347
f(8)=1
f(9)=31
f(10)=17
f(11)=197
f(12)=73
f(13)=1
f(14)=53
f(15)=1
f(16)=941
f(17)=113
f(18)=1
f(19)=47
f(20)=419
f(21)=149
f(22)=1427
f(23)=101
f(24)=107
f(25)=1697
f(26)=199
f(27)=1
f(28)=397
f(29)=139
f(30)=1
f(31)=1
f(32)=1
f(33)=167
f(34)=523
f(35)=1
f(36)=947
f(37)=2957
f(38)=1
f(39)=71
f(40)=1
f(41)=1
f(42)=1
f(43)=739
f(44)=1
f(45)=1319
f(46)=4091
f(47)=1409
f(48)=1
f(49)=1
f(50)=1549
f(51)=1597
f(52)=4937
f(53)=1
f(54)=349
f(55)=5387
f(56)=1847
f(57)=211
f(58)=1171
f(59)=401
f(60)=1
f(61)=373
f(62)=241
f(63)=89
f(64)=1
f(65)=2339
f(66)=1
f(67)=1
f(68)=503
f(69)=103
f(70)=7907
f(71)=1
f(72)=1
f(73)=1693
f(74)=577
f(75)=983
f(76)=9041
f(77)=3079
f(78)=1
f(79)=1
f(80)=1093
f(81)=3347
f(82)=10247
f(83)=1
f(84)=1
f(85)=1
f(86)=3697
f(87)=3769
f(88)=461
f(89)=1
f(90)=3989
f(91)=1
f(92)=4139
f(93)=281
f(94)=1
f(95)=257
f(96)=4447
f(97)=13577
f(98)=307
f(99)=937

b) Substitution of the polynom
The polynom f(x)=x^2+43x-3  could be written as f(y)= y^2-465.25 with x=y-21.5

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+21.5
f'(x)>2x+42

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, 41, 29, 5, 37, 79, 97, 347, 1, 31, 17, 197, 73, 1, 53, 1, 941, 113, 1, 47,
Found in Database : 3, 41, 29, 5, 37, 79, 97, 347, 31, 17, 197, 73, 53, 941, 113, 47, 419, 149, 1427, 101, 107, 1697, 199, 397, 139, 167, 523, 947, 2957, 71,
Found in Database : 3, 5, 17, 29, 31, 37, 41, 47, 53, 71, 73, 79, 89, 97, 101, 103, 107, 113, 131, 139, 149,