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Development of Algorithmic Constructions

08:55:52 20.Apr 2024

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Polynom = x^2+28x+67

1. Algorithm

a:=1;
b:=28;
c:=67;


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+28x+67

f(0)=67
f(1)=3
f(2)=127
f(3)=5
f(4)=13
f(5)=29
f(6)=271
f(7)=1
f(8)=71
f(9)=1
f(10)=149
f(11)=31
f(12)=547
f(13)=1
f(14)=131
f(15)=89
f(16)=257
f(17)=1
f(18)=179
f(19)=1
f(20)=79
f(21)=137
f(22)=389
f(23)=1
f(24)=263
f(25)=1
f(26)=1471
f(27)=97
f(28)=109
f(29)=43
f(30)=139
f(31)=1
f(32)=1987
f(33)=1
f(34)=1
f(35)=1
f(36)=2371
f(37)=103
f(38)=1
f(39)=1
f(40)=929
f(41)=181
f(42)=1
f(43)=1
f(44)=647
f(45)=419
f(46)=1
f(47)=449
f(48)=743
f(49)=1
f(50)=3967
f(51)=1
f(52)=1409
f(53)=1
f(54)=1
f(55)=193
f(56)=367
f(57)=307
f(58)=337
f(59)=1
f(60)=5347
f(61)=229
f(62)=5647
f(63)=1
f(64)=397
f(65)=191
f(66)=6271
f(67)=1
f(68)=1319
f(69)=1
f(70)=2309
f(71)=887
f(72)=1
f(73)=1
f(74)=1523
f(75)=487
f(76)=2657
f(77)=1019
f(78)=1667
f(79)=1
f(80)=8707
f(81)=1
f(82)=233
f(83)=1
f(84)=379
f(85)=1
f(86)=9871
f(87)=1259
f(88)=1
f(89)=1
f(90)=10687
f(91)=227
f(92)=383
f(93)=283
f(94)=769
f(95)=113
f(96)=11971
f(97)=1
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+28x+67  could be written as f(y)= y^2-129 with x=y-14

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+14
f'(x)>2x+27

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 : 67, 3, 127, 5, 13, 29, 271, 1, 71, 1, 149, 31, 547, 1, 131, 89, 257, 1, 179, 1,
Found in Database : 67, 3, 127, 5, 13, 29, 271, 71, 149, 31, 547, 131, 89, 257, 179, 79, 137, 389, 263, 1471, 97, 109, 43, 139, 1987, 2371, 103,
Found in Database : 3, 5, 13, 29, 31, 43, 67, 71, 79, 89, 97, 103, 109, 113, 127, 131, 137, 139, 149,