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Development of Algorithmic Constructions

00:00:06 20.Apr 2024

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Polynom = x^2+32x-79

1. Algorithm

a:=1;
b:=32;
c:=-79;


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-79

f(0)=79
f(1)=23
f(2)=11
f(3)=13
f(4)=5
f(5)=53
f(6)=149
f(7)=97
f(8)=241
f(9)=29
f(10)=31
f(11)=197
f(12)=449
f(13)=1
f(14)=113
f(15)=313
f(16)=1
f(17)=1
f(18)=821
f(19)=89
f(20)=1
f(21)=47
f(22)=1109
f(23)=593
f(24)=1
f(25)=673
f(26)=1429
f(27)=757
f(28)=1601
f(29)=1
f(30)=137
f(31)=937
f(32)=179
f(33)=1033
f(34)=433
f(35)=103
f(36)=1
f(37)=1237
f(38)=1
f(39)=269
f(40)=2801
f(41)=1
f(42)=233
f(43)=1
f(44)=653
f(45)=1693
f(46)=1
f(47)=1
f(48)=3761
f(49)=389
f(50)=4021
f(51)=67
f(52)=4289
f(53)=2213
f(54)=83
f(55)=181
f(56)=373
f(57)=227
f(58)=1
f(59)=1
f(60)=5441
f(61)=2797
f(62)=5749
f(63)=2953
f(64)=1213
f(65)=283
f(66)=6389
f(67)=1
f(68)=1
f(69)=1
f(70)=307
f(71)=3617
f(72)=239
f(73)=3793
f(74)=1553
f(75)=1
f(76)=739
f(77)=4157
f(78)=8501
f(79)=1
f(80)=107
f(81)=349
f(82)=1
f(83)=4733
f(84)=1933
f(85)=4933
f(86)=10069
f(87)=467
f(88)=223
f(89)=1069
f(90)=991
f(91)=5557
f(92)=11329
f(93)=251
f(94)=1
f(95)=461
f(96)=421
f(97)=6217
f(98)=1151
f(99)=1289

b) Substitution of the polynom
The polynom f(x)=x^2+32x-79  could be written as f(y)= y^2-335 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+31

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 : 79, 23, 11, 13, 5, 53, 149, 97, 241, 29, 31, 197, 449, 1, 113, 313, 1, 1, 821, 89,
Found in Database : 79, 23, 11, 13, 5, 53, 149, 97, 241, 29, 31, 197, 449, 113, 313, 821, 89, 47, 1109, 593, 673, 1429, 757, 1601, 137, 937, 179, 1033, 433, 103, 1237, 269,
Found in Database : 5, 11, 13, 23, 29, 31, 47, 53, 67, 79, 83, 89, 97, 103, 107, 113, 127, 137, 139, 149,