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Development of Algorithmic Constructions

11:21:38 19.Apr 2024

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Polynom = x^2-148x+467

1. Algorithm

a:=1;
b:=-148;
c:=467;


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^2-148x+467

f(0)=467
f(1)=5
f(2)=7
f(3)=1
f(4)=109
f(5)=31
f(6)=11
f(7)=13
f(8)=653
f(9)=1
f(10)=83
f(11)=1
f(12)=233
f(13)=23
f(14)=1409
f(15)=191
f(16)=47
f(17)=1
f(18)=1873
f(19)=1
f(20)=1
f(21)=1
f(22)=461
f(23)=43
f(24)=193
f(25)=163
f(26)=541
f(27)=1
f(28)=263
f(29)=373
f(30)=439
f(31)=79
f(32)=59
f(33)=1
f(34)=487
f(35)=1
f(36)=1
f(37)=1
f(38)=1
f(39)=1
f(40)=3853
f(41)=1
f(42)=797
f(43)=1
f(44)=587
f(45)=521
f(46)=1
f(47)=107
f(48)=619
f(49)=137
f(50)=1
f(51)=1
f(52)=181
f(53)=571
f(54)=419
f(55)=1
f(56)=937
f(57)=1
f(58)=97
f(59)=1
f(60)=4813
f(61)=1
f(62)=139
f(63)=1
f(64)=4909
f(65)=1
f(66)=1
f(67)=1
f(68)=4973
f(69)=89
f(70)=4993
f(71)=1
f(72)=1
f(73)=313
f(74)=5009
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^2-148x+467  could be written as f(y)= y^2-5009 with x=y+74

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-74
f'(x)>2x-149  with x > 71

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 : 467, 5, 7, 1, 109, 31, 11, 13, 653, 1, 83, 1, 233, 23, 1409, 191, 47, 1, 1873, 1,
Found in Database : 467, 5, 7, 109, 31, 11, 13, 653, 83, 233, 23, 1409, 191, 47, 1873, 461, 43, 193, 163, 541, 263, 373, 439, 79, 59, 487,
Found in Database : 5, 7, 11, 13, 23, 31, 43, 47, 59, 79, 83, 89, 97, 107, 109, 137, 139,