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Development of Algorithmic Constructions

08:47:12 19.Apr 2024

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Polynom = x^2+82x-3

1. Algorithm

a:=1;
b:=82;
c:=-3;


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+82x-3

f(0)=3
f(1)=5
f(2)=11
f(3)=7
f(4)=31
f(5)=1
f(6)=1
f(7)=1
f(8)=239
f(9)=17
f(10)=131
f(11)=1
f(12)=1
f(13)=1
f(14)=149
f(15)=1
f(16)=313
f(17)=1
f(18)=599
f(19)=479
f(20)=97
f(21)=1
f(22)=457
f(23)=67
f(24)=1
f(25)=167
f(26)=1
f(27)=1
f(28)=181
f(29)=1
f(30)=373
f(31)=1
f(32)=1
f(33)=79
f(34)=563
f(35)=1
f(36)=283
f(37)=1
f(38)=1
f(39)=1
f(40)=4877
f(41)=1
f(42)=347
f(43)=1
f(44)=1847
f(45)=1
f(46)=107
f(47)=101
f(48)=1
f(49)=401
f(50)=733
f(51)=113
f(52)=199
f(53)=1
f(54)=2447
f(55)=269
f(56)=103
f(57)=1
f(58)=8117
f(59)=1
f(60)=1
f(61)=109
f(62)=1
f(63)=761
f(64)=9341
f(65)=1
f(66)=1
f(67)=499
f(68)=1
f(69)=1
f(70)=967
f(71)=1
f(72)=739
f(73)=1
f(74)=3847
f(75)=1
f(76)=1
f(77)=1
f(78)=4159
f(79)=1
f(80)=617
f(81)=1
f(82)=2689
f(83)=163
f(84)=1549
f(85)=887
f(86)=1
f(87)=1
f(88)=14957
f(89)=317
f(90)=1
f(91)=787
f(92)=1
f(93)=1
f(94)=139
f(95)=467
f(96)=1
f(97)=1
f(98)=5879
f(99)=1493

b) Substitution of the polynom
The polynom f(x)=x^2+82x-3  could be written as f(y)= y^2-1684 with x=y-41

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+41
f'(x)>2x+81

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 : 3, 5, 11, 7, 31, 1, 1, 1, 239, 17, 131, 1, 1, 1, 149, 1, 313, 1, 599, 479,
Found in Database : 3, 5, 11, 7, 31, 239, 17, 131, 149, 313, 599, 479, 97, 457, 67, 167, 181, 373, 79, 563, 283,
Found in Database : 3, 5, 7, 11, 17, 31, 67, 79, 97, 101, 103, 107, 109, 113, 131, 139, 149,