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Development of Algorithmic Constructions

12:32:42 19.Apr 2024

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Polynom = x^2-57x-17

1. Algorithm

a:=1;
b:=-57;
c:=-17;


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^2-57x-17

f(0)=17
f(1)=73
f(2)=127
f(3)=179
f(4)=229
f(5)=277
f(6)=19
f(7)=367
f(8)=409
f(9)=449
f(10)=487
f(11)=523
f(12)=557
f(13)=31
f(14)=619
f(15)=647
f(16)=673
f(17)=41
f(18)=719
f(19)=739
f(20)=757
f(21)=773
f(22)=787
f(23)=47
f(24)=809
f(25)=43
f(26)=823
f(27)=827
f(28)=829
f(29)=1
f(30)=1
f(31)=1
f(32)=1
f(33)=1
f(34)=1
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)=101
f(60)=163
f(61)=227
f(62)=293
f(63)=1
f(64)=431
f(65)=503
f(66)=577
f(67)=653
f(68)=1
f(69)=811
f(70)=1
f(71)=977
f(72)=1063
f(73)=1151
f(74)=1
f(75)=1
f(76)=1427
f(77)=1523
f(78)=1621
f(79)=1721
f(80)=1823
f(81)=1
f(82)=107
f(83)=2141
f(84)=2251
f(85)=139
f(86)=2477
f(87)=2593
f(88)=2711
f(89)=149
f(90)=2953
f(91)=181
f(92)=3203
f(93)=3331
f(94)=3461
f(95)=3593
f(96)=3727
f(97)=3863
f(98)=4001
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-57x-17  could be written as f(y)= y^2-829.25 with x=y+28.5

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-28.5
f'(x)>2x-58  with x > 29

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 : 17, 73, 127, 179, 229, 277, 19, 367, 409, 449, 487, 523, 557, 31, 619, 647, 673, 41, 719, 739,
Found in Database : 17, 73, 127, 179, 229, 277, 19, 367, 409, 449, 487, 523, 557, 31, 619, 647, 673, 41, 719, 739, 757, 773, 787, 47, 809, 43, 823, 827, 829,
Found in Database : 17, 19, 31, 41, 43, 47, 73, 101, 107, 127, 139, 149,