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Development of Algorithmic Constructions

07:40:39 20.Apr 2024

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Polynom = x^2+9x-17

1. Algorithm

a:=1;
b:=9;
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+9x-17

f(0)=17
f(1)=7
f(2)=5
f(3)=19
f(4)=1
f(5)=53
f(6)=73
f(7)=1
f(8)=1
f(9)=29
f(10)=173
f(11)=1
f(12)=47
f(13)=269
f(14)=61
f(15)=1
f(16)=383
f(17)=1
f(18)=67
f(19)=103
f(20)=563
f(21)=613
f(22)=1
f(23)=719
f(24)=31
f(25)=1
f(26)=1
f(27)=191
f(28)=1019
f(29)=1
f(30)=1153
f(31)=1223
f(32)=37
f(33)=1
f(34)=1
f(35)=1523
f(36)=229
f(37)=337
f(38)=1
f(39)=1
f(40)=1
f(41)=107
f(42)=1
f(43)=317
f(44)=463
f(45)=127
f(46)=359
f(47)=523
f(48)=2719
f(49)=113
f(50)=419
f(51)=179
f(52)=631
f(53)=467
f(54)=677
f(55)=1
f(56)=3623
f(57)=1
f(58)=1
f(59)=1
f(60)=1
f(61)=4253
f(62)=877
f(63)=4519
f(64)=1
f(65)=4793
f(66)=4933
f(67)=1
f(68)=307
f(69)=1
f(70)=149
f(71)=809
f(72)=1163
f(73)=1
f(74)=1
f(75)=1
f(76)=379
f(77)=1321
f(78)=967
f(79)=1
f(80)=7103
f(81)=1039
f(82)=1489
f(83)=401
f(84)=1559
f(85)=1
f(86)=263
f(87)=1667
f(88)=1217
f(89)=1741
f(90)=8893
f(91)=293
f(92)=1
f(93)=557
f(94)=1933
f(95)=1409
f(96)=347
f(97)=2053
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+9x-17  could be written as f(y)= y^2-37.25 with x=y-4.5

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+4.5
f'(x)>2x+8

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, 7, 5, 19, 1, 53, 73, 1, 1, 29, 173, 1, 47, 269, 61, 1, 383, 1, 67, 103,
Found in Database : 17, 7, 5, 19, 53, 73, 29, 173, 47, 269, 61, 383, 67, 103, 563, 613, 719, 31, 191, 1019, 1153, 1223, 37, 1523, 229, 337,
Found in Database : 5, 7, 17, 19, 29, 31, 37, 47, 53, 61, 67, 73, 103, 107, 113, 127, 149,