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Development of Algorithmic Constructions

19:03:42 18.Apr 2024

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Polynom = x^2-68x+11

1. Algorithm

a:=1;
b:=-68;
c:=11;


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^2-68x+11

f(0)=11
f(1)=7
f(2)=1
f(3)=23
f(4)=5
f(5)=19
f(6)=1
f(7)=13
f(8)=67
f(9)=1
f(10)=569
f(11)=1
f(12)=661
f(13)=1
f(14)=149
f(15)=1
f(16)=821
f(17)=107
f(18)=127
f(19)=1
f(20)=73
f(21)=61
f(22)=1
f(23)=1
f(24)=1
f(25)=1
f(26)=47
f(27)=137
f(28)=1109
f(29)=1
f(30)=1129
f(31)=71
f(32)=163
f(33)=1
f(34)=229
f(35)=1
f(36)=1
f(37)=1
f(38)=1
f(39)=1
f(40)=1
f(41)=1
f(42)=1
f(43)=1
f(44)=1
f(45)=1
f(46)=1
f(47)=1
f(48)=1
f(49)=1
f(50)=1
f(51)=1
f(52)=1
f(53)=1
f(54)=1
f(55)=1
f(56)=1
f(57)=1
f(58)=1
f(59)=1
f(60)=1
f(61)=1
f(62)=1
f(63)=1
f(64)=1
f(65)=1
f(66)=1
f(67)=1
f(68)=1
f(69)=1
f(70)=151
f(71)=1
f(72)=1
f(73)=1
f(74)=1
f(75)=1
f(76)=619
f(77)=1
f(78)=113
f(79)=1
f(80)=971
f(81)=1
f(82)=1
f(83)=157
f(84)=271
f(85)=1
f(86)=1559
f(87)=1
f(88)=1
f(89)=1
f(90)=181
f(91)=263
f(92)=317
f(93)=1
f(94)=491
f(95)=1
f(96)=2699
f(97)=353
f(98)=227
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-68x+11  could be written as f(y)= y^2-1145 with x=y+34

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-34
f'(x)>2x-69  with x > 34

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 : 11, 7, 1, 23, 5, 19, 1, 13, 67, 1, 569, 1, 661, 1, 149, 1, 821, 107, 127, 1,
Found in Database : 11, 7, 23, 5, 19, 13, 67, 569, 661, 149, 821, 107, 127, 73, 61, 47, 137, 1109, 1129, 71, 163, 229,
Found in Database : 5, 7, 11, 13, 19, 23, 47, 61, 67, 71, 73, 107, 113, 127, 137, 149,