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Development of Algorithmic Constructions

15:10:54 20.Apr 2024

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Polynom = x^2-63x-31

1. Algorithm

a:=1;
b:=-63;
c:=-31;


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^2-63x-31

f(0)=31
f(1)=3
f(2)=17
f(3)=211
f(4)=89
f(5)=107
f(6)=373
f(7)=47
f(8)=157
f(9)=11
f(10)=1
f(11)=67
f(12)=643
f(13)=227
f(14)=239
f(15)=751
f(16)=29
f(17)=271
f(18)=1
f(19)=1
f(20)=1
f(21)=83
f(22)=311
f(23)=317
f(24)=967
f(25)=109
f(26)=331
f(27)=59
f(28)=337
f(29)=113
f(30)=1021
f(31)=1
f(32)=1
f(33)=1
f(34)=1
f(35)=1
f(36)=1
f(37)=1
f(38)=1
f(39)=1
f(40)=1
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)=167
f(67)=79
f(68)=103
f(69)=383
f(70)=1
f(71)=179
f(72)=617
f(73)=233
f(74)=1
f(75)=1
f(76)=1
f(77)=349
f(78)=1
f(79)=137
f(80)=443
f(81)=1427
f(82)=509
f(83)=181
f(84)=1733
f(85)=613
f(86)=1
f(87)=1
f(88)=241
f(89)=761
f(90)=2399
f(91)=839
f(92)=293
f(93)=1
f(94)=1
f(95)=1
f(96)=3137
f(97)=1
f(98)=1
f(99)=3533

b) Substitution of the polynom
The polynom f(x)=x^2-63x-31  could be written as f(y)= y^2-1023.25 with x=y+31.5

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-31.5
f'(x)>2x-64  with x > 32

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 : 31, 3, 17, 211, 89, 107, 373, 47, 157, 11, 1, 67, 643, 227, 239, 751, 29, 271, 1, 1,
Found in Database : 31, 3, 17, 211, 89, 107, 373, 47, 157, 11, 67, 643, 227, 239, 751, 29, 271, 83, 311, 317, 967, 109, 331, 59, 337, 113, 1021,
Found in Database : 3, 11, 17, 29, 31, 47, 59, 67, 79, 83, 89, 103, 107, 109, 113, 137,