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Development of Algorithmic Constructions

06:53:54 20.Apr 2024

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Polynom = x^2+33x-101

1. Algorithm

a:=1;
b:=33;
c:=-101;


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+33x-101

f(0)=101
f(1)=67
f(2)=31
f(3)=7
f(4)=47
f(5)=89
f(6)=19
f(7)=179
f(8)=227
f(9)=277
f(10)=1
f(11)=383
f(12)=439
f(13)=71
f(14)=557
f(15)=619
f(16)=683
f(17)=107
f(18)=43
f(19)=887
f(20)=137
f(21)=1033
f(22)=1109
f(23)=1187
f(24)=181
f(25)=1
f(26)=1433
f(27)=1
f(28)=1607
f(29)=1697
f(30)=1789
f(31)=269
f(32)=1979
f(33)=1
f(34)=311
f(35)=53
f(36)=2383
f(37)=131
f(38)=1
f(39)=2707
f(40)=2819
f(41)=419
f(42)=3049
f(43)=3167
f(44)=173
f(45)=487
f(46)=3533
f(47)=3659
f(48)=541
f(49)=3917
f(50)=4049
f(51)=1
f(52)=617
f(53)=4457
f(54)=4597
f(55)=677
f(56)=257
f(57)=1
f(58)=167
f(59)=761
f(60)=5479
f(61)=1
f(62)=827
f(63)=313
f(64)=197
f(65)=6269
f(66)=919
f(67)=6599
f(68)=1
f(69)=991
f(70)=7109
f(71)=7283
f(72)=7459
f(73)=1091
f(74)=7817
f(75)=421
f(76)=1
f(77)=8369
f(78)=199
f(79)=8747
f(80)=1277
f(81)=9133
f(82)=491
f(83)=1361
f(84)=1
f(85)=9929
f(86)=10133
f(87)=211
f(88)=1
f(89)=347
f(90)=1567
f(91)=1
f(92)=11399
f(93)=11617
f(94)=1
f(95)=389
f(96)=1
f(97)=1787
f(98)=271
f(99)=12967

b) Substitution of the polynom
The polynom f(x)=x^2+33x-101  could be written as f(y)= y^2-373.25 with x=y-16.5

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+16.5
f'(x)>2x+32

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 : 101, 67, 31, 7, 47, 89, 19, 179, 227, 277, 1, 383, 439, 71, 557, 619, 683, 107, 43, 887,
Found in Database : 101, 67, 31, 7, 47, 89, 19, 179, 227, 277, 383, 439, 71, 557, 619, 683, 107, 43, 887, 137, 1033, 1109, 1187, 181, 1433, 1607, 1697, 1789, 269, 1979, 311, 53, 2383, 131, 2707,
Found in Database : 7, 19, 31, 43, 47, 53, 67, 71, 89, 101, 107, 131, 137,