---

Development of Algorithmic Constructions

17:41:19 18.Apr 2024

---

Polynom = x^2+44x-733

1. Algorithm

a:=1;
b:=44;
c:=-733;


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+44x-733

f(0)=733
f(1)=43
f(2)=641
f(3)=37
f(4)=541
f(5)=61
f(6)=433
f(7)=47
f(8)=317
f(9)=1
f(10)=193
f(11)=1
f(12)=1
f(13)=1
f(14)=79
f(15)=19
f(16)=227
f(17)=1
f(18)=383
f(19)=29
f(20)=547
f(21)=1
f(22)=719
f(23)=101
f(24)=31
f(25)=1
f(26)=1087
f(27)=1
f(28)=1283
f(29)=173
f(30)=1487
f(31)=199
f(32)=1699
f(33)=113
f(34)=1
f(35)=127
f(36)=1
f(37)=283
f(38)=2383
f(39)=313
f(40)=71
f(41)=1
f(42)=2879
f(43)=1
f(44)=73
f(45)=409
f(46)=3407
f(47)=443
f(48)=1
f(49)=239
f(50)=3967
f(51)=257
f(52)=4259
f(53)=1
f(54)=97
f(55)=1
f(56)=157
f(57)=1
f(58)=1
f(59)=167
f(60)=5507
f(61)=709
f(62)=5839
f(63)=751
f(64)=1
f(65)=397
f(66)=107
f(67)=419
f(68)=6883
f(69)=883
f(70)=7247
f(71)=929
f(72)=401
f(73)=1
f(74)=421
f(75)=1
f(76)=8387
f(77)=1
f(78)=8783
f(79)=1123
f(80)=9187
f(81)=587
f(82)=331
f(83)=613
f(84)=233
f(85)=1279
f(86)=337
f(87)=1
f(88)=10883
f(89)=347
f(90)=241
f(91)=1
f(92)=11779
f(93)=1
f(94)=12239
f(95)=1559
f(96)=131
f(97)=809
f(98)=13183
f(99)=839

b) Substitution of the polynom
The polynom f(x)=x^2+44x-733  could be written as f(y)= y^2-1217 with x=y-22

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+22
f'(x)>2x+43

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 : 733, 43, 641, 37, 541, 61, 433, 47, 317, 1, 193, 1, 1, 1, 79, 19, 227, 1, 383, 29,
Found in Database : 733, 43, 641, 37, 541, 61, 433, 47, 317, 193, 79, 19, 227, 383, 29, 547, 719, 101, 31, 1087, 1283, 173, 1487, 199, 1699, 113, 127, 283, 2383, 313,
Found in Database : 19, 29, 31, 37, 43, 47, 61, 71, 73, 79, 97, 101, 107, 113, 127, 131, 137, 149,