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Development of Algorithmic Constructions

19:28:14 19.Apr 2024

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Polynom = x^2-128x-7

1. Algorithm

a:=1;
b:=-128;
c:=-7;


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^2-128x-7

f(0)=7
f(1)=67
f(2)=37
f(3)=191
f(4)=503
f(5)=311
f(6)=739
f(7)=61
f(8)=967
f(9)=11
f(10)=1187
f(11)=647
f(12)=1399
f(13)=751
f(14)=229
f(15)=23
f(16)=257
f(17)=947
f(18)=1987
f(19)=1039
f(20)=197
f(21)=1
f(22)=2339
f(23)=173
f(24)=2503
f(25)=1291
f(26)=2659
f(27)=1367
f(28)=401
f(29)=1439
f(30)=421
f(31)=137
f(32)=3079
f(33)=1571
f(34)=3203
f(35)=233
f(36)=3319
f(37)=241
f(38)=149
f(39)=47
f(40)=3527
f(41)=1787
f(42)=1
f(43)=1831
f(44)=1
f(45)=1871
f(46)=3779
f(47)=1907
f(48)=3847
f(49)=277
f(50)=3907
f(51)=281
f(52)=107
f(53)=181
f(54)=4003
f(55)=2011
f(56)=577
f(57)=2027
f(58)=83
f(59)=2039
f(60)=1
f(61)=89
f(62)=4099
f(63)=293
f(64)=373
f(65)=1
f(66)=1
f(67)=1
f(68)=1
f(69)=1
f(70)=1
f(71)=1
f(72)=1
f(73)=1
f(74)=1
f(75)=1
f(76)=1
f(77)=1
f(78)=1
f(79)=1
f(80)=1
f(81)=1
f(82)=1
f(83)=1
f(84)=1
f(85)=1
f(86)=1
f(87)=1
f(88)=1
f(89)=1
f(90)=1
f(91)=1
f(92)=1
f(93)=1
f(94)=1
f(95)=1
f(96)=1
f(97)=1
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-128x-7  could be written as f(y)= y^2-4103 with x=y+64

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-64
f'(x)>2x-129  with x > 64

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 : 7, 67, 37, 191, 503, 311, 739, 61, 967, 11, 1187, 647, 1399, 751, 229, 23, 257, 947, 1987, 1039,
Found in Database : 7, 67, 37, 191, 503, 311, 739, 61, 967, 11, 1187, 647, 1399, 751, 229, 23, 257, 947, 1987, 1039, 197, 2339, 173, 2503, 1291, 2659, 1367, 401, 1439, 421, 137, 3079, 1571, 3203, 233, 3319, 241, 149, 47,
Found in Database : 7, 11, 23, 37, 47, 61, 67, 83, 89, 107, 137, 149,