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Development of Algorithmic Constructions

05:14:18 20.Apr 2024

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Polynom = x^2-188x+139

1. Algorithm

a:=1;
b:=-188;
c:=139;


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^2-188x+139

f(0)=139
f(1)=3
f(2)=233
f(3)=13
f(4)=199
f(5)=97
f(6)=953
f(7)=47
f(8)=1301
f(9)=23
f(10)=547
f(11)=113
f(12)=1973
f(13)=89
f(14)=2297
f(15)=307
f(16)=67
f(17)=173
f(18)=127
f(19)=1
f(20)=3221
f(21)=421
f(22)=1171
f(23)=457
f(24)=3797
f(25)=41
f(26)=4073
f(27)=263
f(28)=1447
f(29)=43
f(30)=107
f(31)=197
f(32)=211
f(33)=311
f(34)=1699
f(35)=163
f(36)=5333
f(37)=227
f(38)=83
f(39)=709
f(40)=1
f(41)=1
f(42)=461
f(43)=1
f(44)=6197
f(45)=787
f(46)=2131
f(47)=811
f(48)=6581
f(49)=1
f(50)=6761
f(51)=1
f(52)=2311
f(53)=877
f(54)=151
f(55)=1
f(56)=7253
f(57)=229
f(58)=2467
f(59)=467
f(60)=7541
f(61)=317
f(62)=7673
f(63)=967
f(64)=1
f(65)=491
f(66)=193
f(67)=1
f(68)=617
f(69)=1009
f(70)=2707
f(71)=1021
f(72)=191
f(73)=1
f(74)=8297
f(75)=521
f(76)=2791
f(77)=1051
f(78)=367
f(79)=353
f(80)=8501
f(81)=1
f(82)=2851
f(83)=1
f(84)=8597
f(85)=359
f(86)=1
f(87)=1
f(88)=2887
f(89)=271
f(90)=8681
f(91)=181
f(92)=8693
f(93)=1087
f(94)=223
f(95)=1
f(96)=1
f(97)=1
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-188x+139  could be written as f(y)= y^2-8697 with x=y+94

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-94
f'(x)>2x-189  with x > 93

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 : 139, 3, 233, 13, 199, 97, 953, 47, 1301, 23, 547, 113, 1973, 89, 2297, 307, 67, 173, 127, 1,
Found in Database : 139, 3, 233, 13, 199, 97, 953, 47, 1301, 23, 547, 113, 1973, 89, 2297, 307, 67, 173, 127, 3221, 421, 1171, 457, 3797, 41, 4073, 263, 1447, 43, 107, 197, 211, 311, 1699, 163, 5333, 227, 83, 709,
Found in Database : 3, 13, 23, 41, 43, 47, 67, 83, 89, 97, 107, 113, 127, 139,