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Development of Algorithmic Constructions

04:05:02 20.Apr 2024

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Polynom = x^2+21x+11

1. Algorithm

a:=1;
b:=21;
c:=11;


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+21x+11

f(0)=11
f(1)=3
f(2)=19
f(3)=83
f(4)=37
f(5)=47
f(6)=173
f(7)=23
f(8)=1
f(9)=281
f(10)=107
f(11)=1
f(12)=1
f(13)=151
f(14)=167
f(15)=29
f(16)=67
f(17)=73
f(18)=31
f(19)=257
f(20)=277
f(21)=1
f(22)=1
f(23)=1
f(24)=1091
f(25)=43
f(26)=137
f(27)=1307
f(28)=461
f(29)=487
f(30)=1
f(31)=541
f(32)=569
f(33)=163
f(34)=1
f(35)=1
f(36)=2063
f(37)=719
f(38)=751
f(39)=2351
f(40)=1
f(41)=1
f(42)=2657
f(43)=307
f(44)=1
f(45)=271
f(46)=1031
f(47)=1069
f(48)=3323
f(49)=1
f(50)=1187
f(51)=127
f(52)=1
f(53)=1
f(54)=131
f(55)=1
f(56)=1
f(57)=4457
f(58)=1531
f(59)=1
f(60)=4871
f(61)=557
f(62)=191
f(63)=5303
f(64)=79
f(65)=1867
f(66)=523
f(67)=179
f(68)=1
f(69)=6221
f(70)=709
f(71)=727
f(72)=353
f(73)=1
f(74)=2347
f(75)=7211
f(76)=1
f(77)=229
f(78)=1
f(79)=293
f(80)=1
f(81)=8273
f(82)=2819
f(83)=1
f(84)=8831
f(85)=97
f(86)=1
f(87)=409
f(88)=1
f(89)=1
f(90)=1
f(91)=1
f(92)=3469
f(93)=10613
f(94)=3607
f(95)=3677
f(96)=11243
f(97)=1
f(98)=1297
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+21x+11  could be written as f(y)= y^2-99.25 with x=y-10.5

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+10.5
f'(x)>2x+20

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 : 11, 3, 19, 83, 37, 47, 173, 23, 1, 281, 107, 1, 1, 151, 167, 29, 67, 73, 31, 257,
Found in Database : 11, 3, 19, 83, 37, 47, 173, 23, 281, 107, 151, 167, 29, 67, 73, 31, 257, 277, 1091, 43, 137, 1307, 461, 487, 541, 569, 163, 2063, 719, 751, 2351,
Found in Database : 3, 11, 19, 23, 29, 31, 37, 43, 47, 67, 73, 79, 83, 97, 107, 127, 131, 137,