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Development of Algorithmic Constructions

01:46:49 19.Apr 2024

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Polynom = x^2-138x+131

1. Algorithm

a:=1;
b:=-138;
c:=131;


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^2-138x+131

f(0)=131
f(1)=3
f(2)=47
f(3)=137
f(4)=5
f(5)=89
f(6)=661
f(7)=1
f(8)=101
f(9)=103
f(10)=383
f(11)=211
f(12)=1381
f(13)=83
f(14)=107
f(15)=857
f(16)=607
f(17)=1
f(18)=2029
f(19)=71
f(20)=743
f(21)=1163
f(22)=269
f(23)=419
f(24)=521
f(25)=449
f(26)=1
f(27)=1433
f(28)=983
f(29)=1
f(30)=3109
f(31)=59
f(32)=1087
f(33)=1667
f(34)=227
f(35)=193
f(36)=3541
f(37)=601
f(38)=1223
f(39)=373
f(40)=421
f(41)=641
f(42)=1
f(43)=659
f(44)=1
f(45)=2027
f(46)=1367
f(47)=691
f(48)=1
f(49)=1
f(50)=1423
f(51)=2153
f(52)=1447
f(53)=1
f(54)=881
f(55)=739
f(56)=1487
f(57)=2243
f(58)=167
f(59)=151
f(60)=4549
f(61)=761
f(62)=509
f(63)=2297
f(64)=307
f(65)=769
f(66)=4621
f(67)=257
f(68)=1543
f(69)=463
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^2-138x+131  could be written as f(y)= y^2-4630 with x=y+69

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-69
f'(x)>2x-139  with x > 68

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 : 131, 3, 47, 137, 5, 89, 661, 1, 101, 103, 383, 211, 1381, 83, 107, 857, 607, 1, 2029, 71,
Found in Database : 131, 3, 47, 137, 5, 89, 661, 101, 103, 383, 211, 1381, 83, 107, 857, 607, 2029, 71, 743, 1163, 269, 419, 521, 449, 1433, 983, 3109, 59, 1087, 1667, 227, 193, 3541, 601, 1223, 373,
Found in Database : 3, 5, 47, 59, 71, 83, 89, 101, 103, 107, 131, 137,