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Development of Algorithmic Constructions

08:03:29 20.Apr 2024

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Polynom = x^2-136x+211

1. Algorithm

a:=1;
b:=-136;
c:=211;


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^2-136x+211

f(0)=211
f(1)=19
f(2)=3
f(3)=47
f(4)=317
f(5)=37
f(6)=569
f(7)=173
f(8)=271
f(9)=233
f(10)=1049
f(11)=97
f(12)=1277
f(13)=347
f(14)=499
f(15)=401
f(16)=1709
f(17)=151
f(18)=1913
f(19)=503
f(20)=1
f(21)=29
f(22)=2297
f(23)=199
f(24)=2477
f(25)=641
f(26)=883
f(27)=683
f(28)=1
f(29)=241
f(30)=2969
f(31)=761
f(32)=1039
f(33)=797
f(34)=3257
f(35)=277
f(36)=3389
f(37)=863
f(38)=1171
f(39)=1
f(40)=191
f(41)=307
f(42)=101
f(43)=947
f(44)=1279
f(45)=971
f(46)=3929
f(47)=331
f(48)=4013
f(49)=1013
f(50)=1
f(51)=1031
f(52)=4157
f(53)=349
f(54)=4217
f(55)=1061
f(56)=1423
f(57)=1
f(58)=227
f(59)=1
f(60)=4349
f(61)=1091
f(62)=1459
f(63)=1097
f(64)=4397
f(65)=367
f(66)=4409
f(67)=1103
f(68)=1471
f(69)=1
f(70)=1
f(71)=1
f(72)=1
f(73)=1
f(74)=1
f(75)=1
f(76)=1
f(77)=1
f(78)=1
f(79)=1
f(80)=1
f(81)=1
f(82)=1
f(83)=1
f(84)=1
f(85)=1
f(86)=1
f(87)=1
f(88)=1
f(89)=1
f(90)=1
f(91)=1
f(92)=1
f(93)=1
f(94)=1
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^2-136x+211  could be written as f(y)= y^2-4413 with x=y+68

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-68
f'(x)>2x-137  with x > 66

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 : 211, 19, 3, 47, 317, 37, 569, 173, 271, 233, 1049, 97, 1277, 347, 499, 401, 1709, 151, 1913, 503,
Found in Database : 211, 19, 3, 47, 317, 37, 569, 173, 271, 233, 1049, 97, 1277, 347, 499, 401, 1709, 151, 1913, 503, 29, 2297, 199, 2477, 641, 883, 683, 241, 2969, 761, 1039, 797, 3257, 277, 3389, 863, 1171,
Found in Database : 3, 19, 29, 37, 47, 97, 101,