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Development of Algorithmic Constructions

15:34:30 19.Apr 2024

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Polynom = x^2-188x+59

1. Algorithm

a:=1;
b:=-188;
c:=59;


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+59

f(0)=59
f(1)=1
f(2)=313
f(3)=31
f(4)=677
f(5)=107
f(6)=1033
f(7)=151
f(8)=1381
f(9)=97
f(10)=1721
f(11)=1
f(12)=2053
f(13)=277
f(14)=2377
f(15)=317
f(16)=2693
f(17)=89
f(18)=3001
f(19)=197
f(20)=3301
f(21)=431
f(22)=3593
f(23)=467
f(24)=3877
f(25)=251
f(26)=4153
f(27)=67
f(28)=4421
f(29)=569
f(30)=1
f(31)=601
f(32)=4933
f(33)=79
f(34)=167
f(35)=331
f(36)=5413
f(37)=691
f(38)=5641
f(39)=719
f(40)=5861
f(41)=373
f(42)=6073
f(43)=193
f(44)=6277
f(45)=797
f(46)=6473
f(47)=821
f(48)=6661
f(49)=211
f(50)=6841
f(51)=433
f(52)=7013
f(53)=887
f(54)=7177
f(55)=907
f(56)=7333
f(57)=463
f(58)=7481
f(59)=1
f(60)=7621
f(61)=1
f(62)=7753
f(63)=977
f(64)=7877
f(65)=1
f(66)=7993
f(67)=503
f(68)=8101
f(69)=1019
f(70)=139
f(71)=1031
f(72)=8293
f(73)=521
f(74)=8377
f(75)=263
f(76)=1
f(77)=1061
f(78)=8521
f(79)=1069
f(80)=8581
f(81)=269
f(82)=1
f(83)=541
f(84)=8677
f(85)=1087
f(86)=8713
f(87)=1091
f(88)=8741
f(89)=547
f(90)=8761
f(91)=137
f(92)=283
f(93)=1097
f(94)=131
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+59  could be written as f(y)= y^2-8777 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 > 94

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 : 59, 1, 313, 31, 677, 107, 1033, 151, 1381, 97, 1721, 1, 2053, 277, 2377, 317, 2693, 89, 3001, 197,
Found in Database : 59, 313, 31, 677, 107, 1033, 151, 1381, 97, 1721, 2053, 277, 2377, 317, 2693, 89, 3001, 197, 3301, 431, 3593, 467, 3877, 251, 4153, 67, 4421, 569, 601, 4933, 79, 167, 331, 5413, 691, 5641, 719,
Found in Database : 31, 59, 67, 79, 89, 97, 107, 131, 137, 139,