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Development of Algorithmic Constructions

16:17:35 18.Apr 2024

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Polynom = x^2-167

1. Algorithm

a:=1;
b:=0;
c:=-167;


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^2-167

f(0)=167
f(1)=83
f(2)=163
f(3)=79
f(4)=151
f(5)=71
f(6)=131
f(7)=59
f(8)=103
f(9)=43
f(10)=67
f(11)=23
f(12)=1
f(13)=1
f(14)=29
f(15)=1
f(16)=89
f(17)=61
f(18)=157
f(19)=97
f(20)=233
f(21)=137
f(22)=317
f(23)=181
f(24)=409
f(25)=229
f(26)=509
f(27)=281
f(28)=617
f(29)=337
f(30)=733
f(31)=397
f(32)=857
f(33)=461
f(34)=1
f(35)=1
f(36)=1129
f(37)=601
f(38)=1277
f(39)=677
f(40)=1433
f(41)=757
f(42)=1597
f(43)=1
f(44)=1
f(45)=929
f(46)=1949
f(47)=1021
f(48)=2137
f(49)=1117
f(50)=2333
f(51)=1217
f(52)=1
f(53)=1321
f(54)=2749
f(55)=1429
f(56)=2969
f(57)=1
f(58)=139
f(59)=1657
f(60)=3433
f(61)=1777
f(62)=3677
f(63)=1901
f(64)=3929
f(65)=2029
f(66)=1
f(67)=2161
f(68)=4457
f(69)=2297
f(70)=4733
f(71)=2437
f(72)=173
f(73)=1
f(74)=5309
f(75)=2729
f(76)=1
f(77)=1
f(78)=1
f(79)=3037
f(80)=271
f(81)=1
f(82)=1
f(83)=3361
f(84)=1
f(85)=3529
f(86)=7229
f(87)=3701
f(88)=7577
f(89)=3877
f(90)=7933
f(91)=4057
f(92)=8297
f(93)=4241
f(94)=8669
f(95)=1
f(96)=9049
f(97)=4621
f(98)=9437
f(99)=4817

b) Substitution of the polynom
The polynom f(x)=x^2-167  could be written as f(y)= y^2-167 with x=y+0

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+0
f'(x)>2x-1  with x > 13

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 : 167, 83, 163, 79, 151, 71, 131, 59, 103, 43, 67, 23, 1, 1, 29, 1, 89, 61, 157, 97,
Found in Database : 167, 83, 163, 79, 151, 71, 131, 59, 103, 43, 67, 23, 29, 89, 61, 157, 97, 233, 137, 317, 181, 409, 229, 509, 281, 617, 337, 733, 397, 857, 461, 1129, 601, 1277, 677,
Found in Database : 23, 29, 43, 59, 61, 67, 71, 79, 83, 89, 97, 103, 131, 137, 139,