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Development of Algorithmic Constructions

15:14:04 20.Apr 2024

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Polynom = x^2+138x-163

1. Algorithm

a:=1;
b:=138;
c:=-163;


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-163

f(0)=163
f(1)=3
f(2)=13
f(3)=5
f(4)=1
f(5)=23
f(6)=701
f(7)=71
f(8)=67
f(9)=29
f(10)=439
f(11)=41
f(12)=1637
f(13)=1
f(14)=131
f(15)=1
f(16)=59
f(17)=103
f(18)=1
f(19)=47
f(20)=37
f(21)=397
f(22)=373
f(23)=1
f(24)=149
f(25)=1
f(26)=1367
f(27)=1
f(28)=1
f(29)=1
f(30)=4877
f(31)=1
f(32)=1759
f(33)=137
f(34)=379
f(35)=491
f(36)=6101
f(37)=263
f(38)=1
f(39)=337
f(40)=773
f(41)=1
f(42)=569
f(43)=127
f(44)=523
f(45)=1009
f(46)=2767
f(47)=79
f(48)=1753
f(49)=1
f(50)=3079
f(51)=1
f(52)=1
f(53)=83
f(54)=157
f(55)=1
f(56)=1
f(57)=1
f(58)=1
f(59)=191
f(60)=11717
f(61)=499
f(62)=4079
f(63)=1
f(64)=1
f(65)=181
f(66)=283
f(67)=1
f(68)=1
f(69)=353
f(70)=4799
f(71)=1223
f(72)=14957
f(73)=1
f(74)=1
f(75)=1
f(76)=1789
f(77)=683
f(78)=1
f(79)=1
f(80)=443
f(81)=1
f(82)=101
f(83)=1
f(84)=3697
f(85)=1
f(86)=6367
f(87)=211
f(88)=1
f(89)=167
f(90)=20357
f(91)=1723
f(92)=2333
f(93)=1
f(94)=1
f(95)=1831
f(96)=769
f(97)=1
f(98)=1531
f(99)=233

b) Substitution of the polynom
The polynom f(x)=x^2+138x-163  could be written as f(y)= y^2-4924 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+137

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 : 163, 3, 13, 5, 1, 23, 701, 71, 67, 29, 439, 41, 1637, 1, 131, 1, 59, 103, 1, 47,
Found in Database : 163, 3, 13, 5, 23, 701, 71, 67, 29, 439, 41, 1637, 131, 59, 103, 47, 37, 397, 373, 149, 1367, 4877, 1759, 137, 379, 491, 6101, 263, 337,
Found in Database : 3, 5, 13, 23, 29, 37, 41, 47, 59, 67, 71, 79, 83, 101, 103, 127, 131, 137, 149,