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Development of Algorithmic Constructions

01:14:11 19.Apr 2024

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Polynom = x^2+108x-149

1. Algorithm

a:=1;
b:=108;
c:=-149;


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+108x-149

f(0)=149
f(1)=5
f(2)=71
f(3)=23
f(4)=13
f(5)=1
f(6)=107
f(7)=41
f(8)=19
f(9)=113
f(10)=1031
f(11)=29
f(12)=1291
f(13)=89
f(14)=1559
f(15)=53
f(16)=367
f(17)=1
f(18)=163
f(19)=283
f(20)=2411
f(21)=1
f(22)=2711
f(23)=179
f(24)=3019
f(25)=397
f(26)=1
f(27)=1
f(28)=3659
f(29)=239
f(30)=307
f(31)=1
f(32)=61
f(33)=563
f(34)=4679
f(35)=607
f(36)=1
f(37)=1
f(38)=5399
f(39)=349
f(40)=199
f(41)=1
f(42)=6151
f(43)=1
f(44)=503
f(45)=421
f(46)=73
f(47)=223
f(48)=1
f(49)=1
f(50)=337
f(51)=1
f(52)=8171
f(53)=131
f(54)=8599
f(55)=1
f(56)=139
f(57)=1
f(58)=9479
f(59)=1213
f(60)=9931
f(61)=127
f(62)=10391
f(63)=83
f(64)=10859
f(65)=1
f(66)=2267
f(67)=1447
f(68)=1
f(69)=1
f(70)=947
f(71)=157
f(72)=557
f(73)=1
f(74)=701
f(75)=1697
f(76)=2767
f(77)=881
f(78)=173
f(79)=457
f(80)=14891
f(81)=379
f(82)=1187
f(83)=151
f(84)=1
f(85)=1
f(86)=3307
f(87)=1051
f(88)=17099
f(89)=1
f(90)=431
f(91)=449
f(92)=18251
f(93)=1
f(94)=18839
f(95)=1
f(96)=1
f(97)=2467
f(98)=691
f(99)=2543

b) Substitution of the polynom
The polynom f(x)=x^2+108x-149  could be written as f(y)= y^2-3065 with x=y-54

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+54
f'(x)>2x+107

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 : 149, 5, 71, 23, 13, 1, 107, 41, 19, 113, 1031, 29, 1291, 89, 1559, 53, 367, 1, 163, 283,
Found in Database : 149, 5, 71, 23, 13, 107, 41, 19, 113, 1031, 29, 1291, 89, 1559, 53, 367, 163, 283, 2411, 2711, 179, 3019, 397, 3659, 239, 307, 61, 563, 4679, 607, 5399, 349,
Found in Database : 5, 13, 19, 23, 29, 41, 53, 61, 71, 73, 83, 89, 107, 113, 127, 131, 139, 149,