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Development of Algorithmic Constructions

08:56:42 19.Apr 2024

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Polynom = x^2+13x+3

1. Algorithm

a:=1;
b:=13;
c:=3;


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+13x+3

f(0)=3
f(1)=17
f(2)=11
f(3)=1
f(4)=71
f(5)=31
f(6)=13
f(7)=1
f(8)=19
f(9)=67
f(10)=233
f(11)=89
f(12)=101
f(13)=1
f(14)=127
f(15)=47
f(16)=467
f(17)=1
f(18)=1
f(19)=1
f(20)=1
f(21)=239
f(22)=773
f(23)=277
f(24)=1
f(25)=953
f(26)=113
f(27)=1
f(28)=1151
f(29)=37
f(30)=431
f(31)=1367
f(32)=1
f(33)=1
f(34)=1601
f(35)=1
f(36)=1
f(37)=109
f(38)=647
f(39)=677
f(40)=193
f(41)=739
f(42)=257
f(43)=2411
f(44)=1
f(45)=1
f(46)=1
f(47)=941
f(48)=977
f(49)=3041
f(50)=1051
f(51)=1
f(52)=199
f(53)=389
f(54)=1
f(55)=197
f(56)=1289
f(57)=1
f(58)=317
f(59)=1
f(60)=487
f(61)=4517
f(62)=1
f(63)=1597
f(64)=4931
f(65)=1
f(66)=1
f(67)=173
f(68)=167
f(69)=1
f(70)=5813
f(71)=1
f(72)=157
f(73)=571
f(74)=1
f(75)=1
f(76)=1
f(77)=2311
f(78)=263
f(79)=661
f(80)=827
f(81)=2539
f(82)=7793
f(83)=2657
f(84)=1
f(85)=641
f(86)=1
f(87)=967
f(88)=523
f(89)=1009
f(90)=281
f(91)=9467
f(92)=3221
f(93)=1
f(94)=10061
f(95)=311
f(96)=1163
f(97)=821
f(98)=1
f(99)=3697

b) Substitution of the polynom
The polynom f(x)=x^2+13x+3  could be written as f(y)= y^2-39.25 with x=y-6.5

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+6.5
f'(x)>2x+12

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 : 3, 17, 11, 1, 71, 31, 13, 1, 19, 67, 233, 89, 101, 1, 127, 47, 467, 1, 1, 1,
Found in Database : 3, 17, 11, 71, 31, 13, 19, 67, 233, 89, 101, 127, 47, 467, 239, 773, 277, 953, 113, 1151, 37, 431, 1367, 1601, 109, 647, 677,
Found in Database : 3, 11, 13, 17, 19, 31, 37, 47, 67, 71, 89, 101, 109, 113, 127,