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Development of Algorithmic Constructions

04:09:28 19.Apr 2024

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Polynom = x^2-20x+563

1. Algorithm

a:=1;
b:=-20;
c:=563;


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^2-20x+563

f(0)=563
f(1)=17
f(2)=31
f(3)=1
f(4)=499
f(5)=61
f(6)=479
f(7)=59
f(8)=467
f(9)=29
f(10)=463
f(11)=1
f(12)=1
f(13)=1
f(14)=1
f(15)=1
f(16)=1
f(17)=1
f(18)=1
f(19)=1
f(20)=1
f(21)=73
f(22)=607
f(23)=79
f(24)=659
f(25)=43
f(26)=719
f(27)=47
f(28)=787
f(29)=103
f(30)=863
f(31)=113
f(32)=947
f(33)=1
f(34)=1039
f(35)=1
f(36)=67
f(37)=149
f(38)=1
f(39)=163
f(40)=1
f(41)=89
f(42)=1487
f(43)=97
f(44)=1619
f(45)=211
f(46)=1759
f(47)=229
f(48)=1907
f(49)=1
f(50)=2063
f(51)=1
f(52)=131
f(53)=1
f(54)=2399
f(55)=311
f(56)=2579
f(57)=167
f(58)=2767
f(59)=179
f(60)=2963
f(61)=383
f(62)=3167
f(63)=409
f(64)=109
f(65)=1
f(66)=1
f(67)=1
f(68)=1
f(69)=1
f(70)=239
f(71)=523
f(72)=1
f(73)=277
f(74)=1
f(75)=293
f(76)=1
f(77)=619
f(78)=5087
f(79)=653
f(80)=173
f(81)=1
f(82)=5647
f(83)=181
f(84)=5939
f(85)=761
f(86)=367
f(87)=1
f(88)=6547
f(89)=419
f(90)=6863
f(91)=439
f(92)=7187
f(93)=919
f(94)=1
f(95)=1
f(96)=271
f(97)=251
f(98)=283
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-20x+563  could be written as f(y)= y^2+463 with x=y+10

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-10
f'(x)>2x-21  with x > 22

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 : 563, 17, 31, 1, 499, 61, 479, 59, 467, 29, 463, 1, 1, 1, 1, 1, 1, 1, 1, 1,
Found in Database : 563, 17, 31, 499, 61, 479, 59, 467, 29, 463, 73, 607, 79, 659, 43, 719, 47, 787, 103, 863, 113, 947, 1039, 67, 149, 163,
Found in Database : 17, 29, 31, 43, 47, 59, 61, 67, 73, 79, 89, 97, 103, 109, 113, 131, 149,