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Development of Algorithmic Constructions

20:02:06 19.Apr 2024

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Polynom = x^2-188x+197

1. Algorithm

a:=1;
b:=-188;
c:=197;


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+197

f(0)=197
f(1)=5
f(2)=7
f(3)=179
f(4)=11
f(5)=359
f(6)=1
f(7)=107
f(8)=113
f(9)=101
f(10)=1583
f(11)=1
f(12)=383
f(13)=1039
f(14)=2239
f(15)=109
f(16)=73
f(17)=271
f(18)=409
f(19)=137
f(20)=3163
f(21)=331
f(22)=691
f(23)=257
f(24)=3739
f(25)=277
f(26)=1
f(27)=83
f(28)=4283
f(29)=2207
f(30)=59
f(31)=467
f(32)=1
f(33)=2459
f(34)=5039
f(35)=2579
f(36)=211
f(37)=1
f(38)=5503
f(39)=401
f(40)=97
f(41)=53
f(42)=1187
f(43)=3019
f(44)=877
f(45)=3119
f(46)=181
f(47)=643
f(48)=593
f(49)=3307
f(50)=6703
f(51)=1
f(52)=1
f(53)=71
f(54)=7039
f(55)=3559
f(56)=1439
f(57)=727
f(58)=1049
f(59)=337
f(60)=1069
f(61)=151
f(62)=1523
f(63)=349
f(64)=1
f(65)=557
f(66)=1571
f(67)=1
f(68)=7963
f(69)=4007
f(70)=733
f(71)=811
f(72)=233
f(73)=4099
f(74)=1
f(75)=4139
f(76)=1663
f(77)=167
f(78)=1
f(79)=601
f(80)=8443
f(81)=1
f(82)=1699
f(83)=4259
f(84)=8539
f(85)=389
f(86)=1
f(87)=859
f(88)=1229
f(89)=1
f(90)=8623
f(91)=863
f(92)=157
f(93)=617
f(94)=163
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+197  could be written as f(y)= y^2-8639 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 : 197, 5, 7, 179, 11, 359, 1, 107, 113, 101, 1583, 1, 383, 1039, 2239, 109, 73, 271, 409, 137,
Found in Database : 197, 5, 7, 179, 11, 359, 107, 113, 101, 1583, 383, 1039, 2239, 109, 73, 271, 409, 137, 3163, 331, 691, 257, 3739, 277, 83, 4283, 2207, 59, 467, 2459, 5039, 2579, 211, 5503, 401,
Found in Database : 5, 7, 11, 53, 59, 71, 73, 83, 97, 101, 107, 109, 113, 137,