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Development of Algorithmic Constructions

16:05:33 19.Apr 2024

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Polynom = x^2+12x+179

1. Algorithm

a:=1;
b:=12;
c:=179;


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+12x+179

f(0)=179
f(1)=3
f(2)=23
f(3)=7
f(4)=1
f(5)=11
f(6)=41
f(7)=13
f(8)=113
f(9)=1
f(10)=19
f(11)=1
f(12)=467
f(13)=1
f(14)=181
f(15)=73
f(16)=1
f(17)=1
f(18)=719
f(19)=1
f(20)=1
f(21)=109
f(22)=103
f(23)=1
f(24)=149
f(25)=1
f(26)=389
f(27)=1
f(28)=433
f(29)=1
f(30)=1439
f(31)=1
f(32)=1
f(33)=1
f(34)=83
f(35)=1
f(36)=1907
f(37)=1
f(38)=1
f(39)=271
f(40)=251
f(41)=1
f(42)=2447
f(43)=53
f(44)=881
f(45)=1
f(46)=1
f(47)=1
f(48)=1
f(49)=1
f(50)=1093
f(51)=1
f(52)=167
f(53)=151
f(54)=197
f(55)=1
f(56)=443
f(57)=257
f(58)=157
f(59)=1
f(60)=409
f(61)=193
f(62)=227
f(63)=613
f(64)=1
f(65)=1
f(66)=761
f(67)=1
f(68)=1873
f(69)=1
f(70)=1973
f(71)=1
f(72)=479
f(73)=1
f(74)=727
f(75)=419
f(76)=1
f(77)=293
f(78)=313
f(79)=307
f(80)=359
f(81)=241
f(82)=239
f(83)=1
f(84)=8243
f(85)=1
f(86)=1
f(87)=1
f(88)=1
f(89)=191
f(90)=1
f(91)=199
f(92)=1
f(93)=1
f(94)=1
f(95)=431
f(96)=1
f(97)=1
f(98)=281
f(99)=349

b) Substitution of the polynom
The polynom f(x)=x^2+12x+179  could be written as f(y)= y^2+143 with x=y-6

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+6
f'(x)>2x+11

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 : 179, 3, 23, 7, 1, 11, 41, 13, 113, 1, 19, 1, 467, 1, 181, 73, 1, 1, 719, 1,
Found in Database : 179, 3, 23, 7, 11, 41, 13, 113, 19, 467, 181, 73, 719, 109, 103, 149, 389, 433, 1439, 83, 1907, 271,
Found in Database : 3, 7, 11, 13, 19, 23, 41, 53, 73, 83, 103, 109, 113, 149,