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Development of Algorithmic Constructions

01:32:03 19.Apr 2024

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Polynom = x^2+96x-17

1. Algorithm

a:=1;
b:=96;
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+96x-17

f(0)=17
f(1)=5
f(2)=179
f(3)=7
f(4)=383
f(5)=61
f(6)=1
f(7)=11
f(8)=163
f(9)=29
f(10)=149
f(11)=1
f(12)=1279
f(13)=1
f(14)=1523
f(15)=103
f(16)=71
f(17)=1
f(18)=37
f(19)=271
f(20)=47
f(21)=1
f(22)=2579
f(23)=1
f(24)=409
f(25)=1
f(26)=631
f(27)=59
f(28)=691
f(29)=41
f(30)=53
f(31)=1
f(32)=4079
f(33)=1
f(34)=1
f(35)=571
f(36)=947
f(37)=613
f(38)=1
f(39)=1
f(40)=1
f(41)=1
f(42)=5779
f(43)=1
f(44)=6143
f(45)=113
f(46)=1303
f(47)=419
f(48)=197
f(49)=443
f(50)=7283
f(51)=1
f(52)=1097
f(53)=1
f(54)=137
f(55)=1
f(56)=1699
f(57)=1
f(58)=1783
f(59)=1
f(60)=9343
f(61)=239
f(62)=127
f(63)=1
f(64)=10223
f(65)=653
f(66)=1
f(67)=1
f(68)=131
f(69)=1
f(70)=283
f(71)=1
f(72)=257
f(73)=1
f(74)=739
f(75)=1601
f(76)=373
f(77)=1663
f(78)=2711
f(79)=863
f(80)=1
f(81)=1
f(82)=1
f(83)=1
f(84)=1373
f(85)=1
f(86)=1
f(87)=1
f(88)=647
f(89)=1
f(90)=2389
f(91)=1
f(92)=467
f(93)=439
f(94)=2549
f(95)=1
f(96)=1
f(97)=167
f(98)=1
f(99)=2411

b) Substitution of the polynom
The polynom f(x)=x^2+96x-17  could be written as f(y)= y^2-2321 with x=y-48

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+48
f'(x)>2x+95

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, 5, 179, 7, 383, 61, 1, 11, 163, 29, 149, 1, 1279, 1, 1523, 103, 71, 1, 37, 271,
Found in Database : 17, 5, 179, 7, 383, 61, 11, 163, 29, 149, 1279, 1523, 103, 71, 37, 271, 47, 2579, 409, 631, 59, 691, 41, 53, 4079, 571, 947, 613,
Found in Database : 5, 7, 11, 17, 29, 37, 41, 47, 53, 59, 61, 71, 103, 113, 127, 131, 137, 149,