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Development of Algorithmic Constructions

13:23:26 20.Apr 2024

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Polynom = x^2-17

1. Algorithm

a:=1;
b:=0;
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^2-17

f(0)=17
f(1)=1
f(2)=13
f(3)=1
f(4)=1
f(5)=1
f(6)=19
f(7)=1
f(8)=47
f(9)=1
f(10)=83
f(11)=1
f(12)=127
f(13)=1
f(14)=179
f(15)=1
f(16)=239
f(17)=1
f(18)=307
f(19)=43
f(20)=383
f(21)=53
f(22)=467
f(23)=1
f(24)=1
f(25)=1
f(26)=659
f(27)=89
f(28)=59
f(29)=103
f(30)=883
f(31)=1
f(32)=1
f(33)=67
f(34)=1
f(35)=151
f(36)=1279
f(37)=1
f(38)=1427
f(39)=1
f(40)=1583
f(41)=1
f(42)=1747
f(43)=229
f(44)=101
f(45)=251
f(46)=2099
f(47)=137
f(48)=2287
f(49)=149
f(50)=191
f(51)=1
f(52)=2687
f(53)=349
f(54)=223
f(55)=1
f(56)=3119
f(57)=1
f(58)=3347
f(59)=433
f(60)=3583
f(61)=463
f(62)=1
f(63)=1
f(64)=4079
f(65)=263
f(66)=4339
f(67)=1
f(68)=271
f(69)=593
f(70)=257
f(71)=157
f(72)=5167
f(73)=1
f(74)=1
f(75)=701
f(76)=443
f(77)=739
f(78)=6067
f(79)=389
f(80)=491
f(81)=409
f(82)=353
f(83)=859
f(84)=7039
f(85)=1
f(86)=1
f(87)=1
f(88)=7727
f(89)=1
f(90)=1
f(91)=1033
f(92)=8447
f(93)=1
f(94)=8819
f(95)=563
f(96)=9199
f(97)=587
f(98)=9587
f(99)=1223

b) Substitution of the polynom
The polynom f(x)=x^2-17  could be written as f(y)= y^2-17 with x=y+0

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+0
f'(x)>2x-1  with x > 4

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, 1, 13, 1, 1, 1, 19, 1, 47, 1, 83, 1, 127, 1, 179, 1, 239, 1, 307, 43,
Found in Database : 17, 13, 19, 47, 83, 127, 179, 239, 307, 43, 383, 53, 467, 659, 89, 59, 103, 883, 67, 151, 1279, 1427,
Found in Database : 13, 17, 19, 43, 47, 53, 59, 67, 83, 89, 101, 103, 127, 137, 149,