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Development of Algorithmic Constructions

22:33:11 19.Apr 2024

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Polynom = x^2-80x-113

1. Algorithm

a:=1;
b:=-80;
c:=-113;


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^2-80x-113

f(0)=113
f(1)=3
f(2)=269
f(3)=43
f(4)=139
f(5)=61
f(6)=557
f(7)=13
f(8)=53
f(9)=47
f(10)=271
f(11)=109
f(12)=929
f(13)=41
f(14)=17
f(15)=1
f(16)=379
f(17)=37
f(18)=1229
f(19)=1
f(20)=101
f(21)=1
f(22)=463
f(23)=89
f(24)=31
f(25)=1
f(26)=1
f(27)=193
f(28)=523
f(29)=199
f(30)=1613
f(31)=1
f(32)=97
f(33)=1
f(34)=1
f(35)=211
f(36)=1697
f(37)=71
f(38)=1709
f(39)=107
f(40)=571
f(41)=1
f(42)=1
f(43)=1
f(44)=1
f(45)=1
f(46)=1
f(47)=1
f(48)=1
f(49)=1
f(50)=1
f(51)=1
f(52)=1
f(53)=1
f(54)=1
f(55)=1
f(56)=1
f(57)=1
f(58)=1
f(59)=1
f(60)=1
f(61)=1
f(62)=1
f(63)=1
f(64)=1
f(65)=1
f(66)=1
f(67)=1
f(68)=1
f(69)=1
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)=223
f(85)=1
f(86)=1
f(87)=1
f(88)=197
f(89)=1
f(90)=787
f(91)=1
f(92)=991
f(93)=137
f(94)=401
f(95)=1
f(96)=1423
f(97)=1
f(98)=127
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-80x-113  could be written as f(y)= y^2-1713 with x=y+40

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-40
f'(x)>2x-81  with x > 41

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 : 113, 3, 269, 43, 139, 61, 557, 13, 53, 47, 271, 109, 929, 41, 17, 1, 379, 37, 1229, 1,
Found in Database : 113, 3, 269, 43, 139, 61, 557, 13, 53, 47, 271, 109, 929, 41, 17, 379, 37, 1229, 101, 463, 89, 31, 193, 523, 199, 1613, 97, 211, 1697, 71, 1709, 107,
Found in Database : 3, 13, 17, 31, 37, 41, 43, 47, 53, 61, 71, 89, 97, 101, 107, 109, 113, 127, 137, 139,