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Development of Algorithmic Constructions

00:44:13 17.Apr 2024

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Polynom = x^2+92x-3

1. Algorithm

a:=1;
b:=92;
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+92x-3

f(0)=3
f(1)=5
f(2)=37
f(3)=47
f(4)=127
f(5)=241
f(6)=13
f(7)=23
f(8)=797
f(9)=151
f(10)=113
f(11)=1
f(12)=83
f(13)=227
f(14)=1481
f(15)=89
f(16)=1
f(17)=1
f(18)=659
f(19)=1
f(20)=2237
f(21)=79
f(22)=167
f(23)=1321
f(24)=103
f(25)=487
f(26)=613
f(27)=107
f(28)=373
f(29)=1753
f(30)=53
f(31)=1
f(32)=61
f(33)=229
f(34)=1427
f(35)=2221
f(36)=307
f(37)=1
f(38)=4937
f(39)=1
f(40)=1759
f(41)=109
f(42)=1
f(43)=967
f(44)=5981
f(45)=1
f(46)=1
f(47)=653
f(48)=2239
f(49)=1151
f(50)=1
f(51)=1
f(52)=499
f(53)=1
f(54)=71
f(55)=449
f(56)=1657
f(57)=283
f(58)=223
f(59)=73
f(60)=1013
f(61)=311
f(62)=1
f(63)=1627
f(64)=1109
f(65)=5101
f(66)=139
f(67)=1
f(68)=149
f(69)=617
f(70)=3779
f(71)=1
f(72)=787
f(73)=1
f(74)=12281
f(75)=2087
f(76)=1
f(77)=1301
f(78)=491
f(79)=2251
f(80)=13757
f(81)=467
f(82)=317
f(83)=137
f(84)=379
f(85)=1
f(86)=3061
f(87)=173
f(88)=5279
f(89)=8053
f(90)=1
f(91)=1
f(92)=677
f(93)=1
f(94)=5827
f(95)=1
f(96)=401
f(97)=1
f(98)=18617
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+92x-3  could be written as f(y)= y^2-2119 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 : 3, 5, 37, 47, 127, 241, 13, 23, 797, 151, 113, 1, 83, 227, 1481, 89, 1, 1, 659, 1,
Found in Database : 3, 5, 37, 47, 127, 241, 13, 23, 797, 151, 113, 83, 227, 1481, 89, 659, 2237, 79, 167, 1321, 103, 487, 613, 107, 373, 1753, 53, 61, 229, 1427, 2221, 307, 4937,
Found in Database : 3, 5, 13, 23, 37, 47, 53, 61, 71, 73, 79, 83, 89, 103, 107, 109, 113, 127, 137, 139, 149,