---

Development of Algorithmic Constructions

08:03:20 18.Apr 2024

---

Polynom = x^2+92x-181

1. Algorithm

a:=1;
b:=92;
c:=-181;


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+92x-181

f(0)=181
f(1)=11
f(2)=7
f(3)=13
f(4)=29
f(5)=19
f(6)=37
f(7)=1
f(8)=619
f(9)=1
f(10)=839
f(11)=17
f(12)=97
f(13)=1
f(14)=1303
f(15)=89
f(16)=1
f(17)=1
f(18)=257
f(19)=241
f(20)=71
f(21)=137
f(22)=179
f(23)=1
f(24)=1
f(25)=1
f(26)=2887
f(27)=379
f(28)=1
f(29)=1
f(30)=1
f(31)=227
f(32)=541
f(33)=1
f(34)=373
f(35)=41
f(36)=233
f(37)=1
f(38)=4759
f(39)=1
f(40)=5099
f(41)=659
f(42)=419
f(43)=1
f(44)=829
f(45)=1
f(46)=881
f(47)=397
f(48)=503
f(49)=1
f(50)=1
f(51)=127
f(52)=7307
f(53)=67
f(54)=7703
f(55)=1
f(56)=1
f(57)=1039
f(58)=1217
f(59)=1091
f(60)=1277
f(61)=1
f(62)=1
f(63)=599
f(64)=9803
f(65)=1
f(66)=10247
f(67)=1
f(68)=823
f(69)=683
f(70)=11159
f(71)=1
f(72)=151
f(73)=1483
f(74)=1
f(75)=1543
f(76)=307
f(77)=401
f(78)=1
f(79)=1
f(80)=367
f(81)=1
f(82)=14087
f(83)=163
f(84)=859
f(85)=929
f(86)=2161
f(87)=1
f(88)=2237
f(89)=1
f(90)=167
f(91)=1
f(92)=16747
f(93)=1
f(94)=1
f(95)=157
f(96)=1051
f(97)=2269
f(98)=18439
f(99)=2341

b) Substitution of the polynom
The polynom f(x)=x^2+92x-181  could be written as f(y)= y^2-2297 with x=y-46

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+46
f'(x)>2x+91

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 : 181, 11, 7, 13, 29, 19, 37, 1, 619, 1, 839, 17, 97, 1, 1303, 89, 1, 1, 257, 241,
Found in Database : 181, 11, 7, 13, 29, 19, 37, 619, 839, 17, 97, 1303, 89, 257, 241, 71, 137, 179, 2887, 379, 227, 541, 373, 41, 233, 4759,
Found in Database : 7, 11, 13, 17, 19, 29, 37, 41, 67, 71, 89, 97, 127, 137,