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Development of Algorithmic Constructions

15:45:06 19.Apr 2024

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Polynom = x^2+104x-1

1. Algorithm

a:=1;
b:=104;
c:=-1;


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+104x-1

f(0)=1
f(1)=13
f(2)=211
f(3)=5
f(4)=431
f(5)=17
f(6)=659
f(7)=97
f(8)=179
f(9)=127
f(10)=67
f(11)=79
f(12)=107
f(13)=19
f(14)=1
f(15)=223
f(16)=101
f(17)=257
f(18)=439
f(19)=73
f(20)=37
f(21)=41
f(22)=163
f(23)=1
f(24)=83
f(25)=31
f(26)=109
f(27)=1
f(28)=739
f(29)=241
f(30)=4019
f(31)=523
f(32)=229
f(33)=113
f(34)=4691
f(35)=1
f(36)=5039
f(37)=1
f(38)=1
f(39)=1
f(40)=443
f(41)=743
f(42)=6131
f(43)=1
f(44)=383
f(45)=419
f(46)=6899
f(47)=887
f(48)=1459
f(49)=937
f(50)=7699
f(51)=1
f(52)=8111
f(53)=1
f(54)=449
f(55)=1093
f(56)=1
f(57)=1
f(58)=1879
f(59)=601
f(60)=9839
f(61)=1
f(62)=251
f(63)=263
f(64)=827
f(65)=1373
f(66)=863
f(67)=1
f(68)=2339
f(69)=373
f(70)=641
f(71)=1553
f(72)=12671
f(73)=1
f(74)=13171
f(75)=839
f(76)=13679
f(77)=1
f(78)=167
f(79)=139
f(80)=359
f(81)=1873
f(82)=151
f(83)=1
f(84)=15791
f(85)=1
f(86)=16339
f(87)=1
f(88)=1
f(89)=1
f(90)=1
f(91)=1109
f(92)=1
f(93)=1
f(94)=503
f(95)=1
f(96)=1
f(97)=2437
f(98)=1
f(99)=157

b) Substitution of the polynom
The polynom f(x)=x^2+104x-1  could be written as f(y)= y^2-2705 with x=y-52

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+52
f'(x)>2x+103

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 : 1, 13, 211, 5, 431, 17, 659, 97, 179, 127, 67, 79, 107, 19, 1, 223, 101, 257, 439, 73,
Found in Database : 13, 211, 5, 431, 17, 659, 97, 179, 127, 67, 79, 107, 19, 223, 101, 257, 439, 73, 37, 41, 163, 83, 31, 109, 739, 241, 4019, 523, 229, 113, 4691, 5039,
Found in Database : 5, 13, 17, 19, 31, 37, 41, 67, 73, 79, 83, 97, 101, 107, 109, 113, 127, 139,