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Development of Algorithmic Constructions

12:38:31 19.Apr 2024

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Polynom = x^2+8x+47

1. Algorithm

a:=1;
b:=8;
c:=47;


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+8x+47

f(0)=47
f(1)=7
f(2)=67
f(3)=5
f(4)=19
f(5)=1
f(6)=131
f(7)=1
f(8)=1
f(9)=1
f(10)=227
f(11)=1
f(12)=41
f(13)=1
f(14)=71
f(15)=1
f(16)=431
f(17)=59
f(18)=103
f(19)=1
f(20)=607
f(21)=1
f(22)=101
f(23)=1
f(24)=163
f(25)=109
f(26)=1
f(27)=31
f(28)=211
f(29)=1
f(30)=1187
f(31)=157
f(32)=1327
f(33)=1
f(34)=1
f(35)=97
f(36)=233
f(37)=107
f(38)=359
f(39)=1
f(40)=281
f(41)=257
f(42)=113
f(43)=1
f(44)=467
f(45)=1
f(46)=2531
f(47)=1
f(48)=547
f(49)=1
f(50)=421
f(51)=191
f(52)=3167
f(53)=1
f(54)=1
f(55)=439
f(56)=3631
f(57)=1
f(58)=1
f(59)=1
f(60)=4127
f(61)=1
f(62)=1
f(63)=1
f(64)=1
f(65)=599
f(66)=4931
f(67)=317
f(68)=149
f(69)=1
f(70)=5507
f(71)=1
f(72)=5807
f(73)=1
f(74)=1223
f(75)=1
f(76)=1
f(77)=1
f(78)=193
f(79)=173
f(80)=373
f(81)=907
f(82)=1061
f(83)=1
f(84)=311
f(85)=1
f(86)=1
f(87)=1039
f(88)=1699
f(89)=1
f(90)=8867
f(91)=283
f(92)=1321
f(93)=1
f(94)=1
f(95)=1229
f(96)=1433
f(97)=1279
f(98)=2087
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+8x+47  could be written as f(y)= y^2+31 with x=y-4

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+4
f'(x)>2x+7

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 : 47, 7, 67, 5, 19, 1, 131, 1, 1, 1, 227, 1, 41, 1, 71, 1, 431, 59, 103, 1,
Found in Database : 47, 7, 67, 5, 19, 131, 227, 41, 71, 431, 59, 103, 607, 101, 163, 109, 31, 211, 1187, 157, 1327, 97, 233, 107, 359,
Found in Database : 5, 7, 19, 31, 41, 47, 59, 67, 71, 97, 101, 103, 107, 109, 113, 131, 149,