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Development of Algorithmic Constructions

09:58:19 20.Apr 2024

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Polynom = x^2+44x-103

1. Algorithm

a:=1;
b:=44;
c:=-103;


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+44x-103

f(0)=103
f(1)=29
f(2)=11
f(3)=19
f(4)=89
f(5)=71
f(6)=197
f(7)=127
f(8)=313
f(9)=17
f(10)=23
f(11)=251
f(12)=569
f(13)=1
f(14)=709
f(15)=1
f(16)=857
f(17)=467
f(18)=1013
f(19)=547
f(20)=107
f(21)=631
f(22)=1
f(23)=719
f(24)=139
f(25)=811
f(26)=101
f(27)=907
f(28)=1913
f(29)=53
f(30)=73
f(31)=1
f(32)=137
f(33)=1
f(34)=2549
f(35)=1
f(36)=2777
f(37)=1447
f(38)=131
f(39)=1567
f(40)=3257
f(41)=1
f(42)=1
f(43)=1
f(44)=3769
f(45)=1951
f(46)=367
f(47)=2087
f(48)=227
f(49)=1
f(50)=4597
f(51)=2371
f(52)=4889
f(53)=229
f(54)=5189
f(55)=2671
f(56)=239
f(57)=257
f(58)=5813
f(59)=1
f(60)=1
f(61)=1
f(62)=6469
f(63)=3319
f(64)=619
f(65)=3491
f(66)=421
f(67)=193
f(68)=683
f(69)=3847
f(70)=7877
f(71)=1
f(72)=113
f(73)=4219
f(74)=8629
f(75)=401
f(76)=1
f(77)=271
f(78)=9413
f(79)=1
f(80)=9817
f(81)=5011
f(82)=1
f(83)=307
f(84)=463
f(85)=5431
f(86)=1
f(87)=5647
f(88)=397
f(89)=5867
f(90)=1087
f(91)=6091
f(92)=12409
f(93)=1
f(94)=757
f(95)=6551
f(96)=13337
f(97)=617
f(98)=727
f(99)=7027

b) Substitution of the polynom
The polynom f(x)=x^2+44x-103  could be written as f(y)= y^2-587 with x=y-22

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+22
f'(x)>2x+43

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 : 103, 29, 11, 19, 89, 71, 197, 127, 313, 17, 23, 251, 569, 1, 709, 1, 857, 467, 1013, 547,
Found in Database : 103, 29, 11, 19, 89, 71, 197, 127, 313, 17, 23, 251, 569, 709, 857, 467, 1013, 547, 107, 631, 719, 139, 811, 101, 907, 1913, 53, 73, 137, 2549, 2777, 1447, 131, 1567,
Found in Database : 11, 17, 19, 23, 29, 53, 71, 73, 89, 101, 103, 107, 113, 127, 131, 137, 139, 149,