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Development of Algorithmic Constructions

18:36:49 19.Apr 2024

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Polynom = x^2+15x-13

1. Algorithm

a:=1;
b:=15;
c:=-13;


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+15x-13

f(0)=13
f(1)=3
f(2)=7
f(3)=41
f(4)=1
f(5)=29
f(6)=113
f(7)=47
f(8)=19
f(9)=1
f(10)=79
f(11)=1
f(12)=311
f(13)=1
f(14)=131
f(15)=23
f(16)=1
f(17)=59
f(18)=83
f(19)=211
f(20)=229
f(21)=743
f(22)=89
f(23)=1
f(24)=71
f(25)=1
f(26)=1
f(27)=1
f(28)=397
f(29)=421
f(30)=191
f(31)=157
f(32)=1
f(33)=1571
f(34)=1
f(35)=193
f(36)=1823
f(37)=1
f(38)=1
f(39)=1
f(40)=1
f(41)=761
f(42)=2381
f(43)=827
f(44)=1
f(45)=2687
f(46)=1
f(47)=967
f(48)=3011
f(49)=347
f(50)=1
f(51)=479
f(52)=1
f(53)=1
f(54)=1
f(55)=1279
f(56)=1321
f(57)=4091
f(58)=67
f(59)=1451
f(60)=641
f(61)=1
f(62)=1
f(63)=1
f(64)=1
f(65)=1
f(66)=5333
f(67)=1
f(68)=1877
f(69)=5783
f(70)=1979
f(71)=677
f(72)=1
f(73)=2137
f(74)=313
f(75)=6737
f(76)=1
f(77)=2357
f(78)=557
f(79)=353
f(80)=281
f(81)=1109
f(82)=2647
f(83)=2707
f(84)=1
f(85)=1
f(86)=1
f(87)=8861
f(88)=431
f(89)=1
f(90)=9437
f(91)=1
f(92)=1
f(93)=1433
f(94)=379
f(95)=1
f(96)=367
f(97)=3617
f(98)=1229
f(99)=11273

b) Substitution of the polynom
The polynom f(x)=x^2+15x-13  could be written as f(y)= y^2-69.25 with x=y-7.5

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+7.5
f'(x)>2x+14

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 : 13, 3, 7, 41, 1, 29, 113, 47, 19, 1, 79, 1, 311, 1, 131, 23, 1, 59, 83, 211,
Found in Database : 13, 3, 7, 41, 29, 113, 47, 19, 79, 311, 131, 23, 59, 83, 211, 229, 743, 89, 71, 397, 421, 191, 157, 1571, 193, 1823,
Found in Database : 3, 7, 13, 19, 23, 29, 41, 47, 59, 67, 71, 79, 83, 89, 113, 131,