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Development of Algorithmic Constructions

01:17:21 19.Apr 2024

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Polynom = x^2-32x+17

1. Algorithm

a:=1;
b:=-32;
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^2-32x+17

f(0)=17
f(1)=7
f(2)=43
f(3)=5
f(4)=19
f(5)=59
f(6)=139
f(7)=79
f(8)=1
f(9)=1
f(10)=29
f(11)=107
f(12)=223
f(13)=23
f(14)=47
f(15)=1
f(16)=239
f(17)=1
f(18)=1
f(19)=1
f(20)=1
f(21)=1
f(22)=1
f(23)=1
f(24)=1
f(25)=1
f(26)=1
f(27)=1
f(28)=1
f(29)=1
f(30)=1
f(31)=1
f(32)=1
f(33)=1
f(34)=1
f(35)=61
f(36)=1
f(37)=101
f(38)=1
f(39)=1
f(40)=337
f(41)=193
f(42)=1
f(43)=1
f(44)=109
f(45)=1
f(46)=661
f(47)=1
f(48)=157
f(49)=1
f(50)=131
f(51)=1
f(52)=151
f(53)=113
f(54)=241
f(55)=641
f(56)=1361
f(57)=103
f(58)=1
f(59)=1
f(60)=1697
f(61)=1
f(62)=1877
f(63)=197
f(64)=1
f(65)=1
f(66)=1
f(67)=1181
f(68)=1
f(69)=257
f(70)=2677
f(71)=199
f(72)=2897
f(73)=1
f(74)=1
f(75)=1621
f(76)=3361
f(77)=1741
f(78)=1
f(79)=373
f(80)=1
f(81)=1993
f(82)=179
f(83)=1
f(84)=877
f(85)=1
f(86)=1
f(87)=1
f(88)=1
f(89)=509
f(90)=5237
f(91)=2693
f(92)=1
f(93)=569
f(94)=167
f(95)=3001
f(96)=1
f(97)=1
f(98)=1297
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-32x+17  could be written as f(y)= y^2-239 with x=y+16

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-16
f'(x)>2x-33  with x > 15

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, 43, 5, 19, 59, 139, 79, 1, 1, 29, 107, 223, 23, 47, 1, 239, 1, 1, 1,
Found in Database : 17, 7, 43, 5, 19, 59, 139, 79, 29, 107, 223, 23, 47, 239, 61, 101,
Found in Database : 5, 7, 17, 19, 23, 29, 43, 47, 59, 61, 79, 101, 103, 107, 109, 113, 131, 139,