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Development of Algorithmic Constructions

10:47:06 19.Apr 2024

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Polynom = x^2+23

1. Algorithm

a:=1;
b:=0;
c:=23;


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+23

f(0)=23
f(1)=3
f(2)=1
f(3)=1
f(4)=13
f(5)=1
f(6)=59
f(7)=1
f(8)=29
f(9)=1
f(10)=41
f(11)=1
f(12)=167
f(13)=1
f(14)=73
f(15)=31
f(16)=1
f(17)=1
f(18)=347
f(19)=1
f(20)=47
f(21)=1
f(22)=1
f(23)=1
f(24)=599
f(25)=1
f(26)=233
f(27)=1
f(28)=269
f(29)=1
f(30)=71
f(31)=1
f(32)=349
f(33)=139
f(34)=131
f(35)=1
f(36)=1319
f(37)=1
f(38)=163
f(39)=193
f(40)=541
f(41)=1
f(42)=1787
f(43)=1
f(44)=653
f(45)=1
f(46)=1
f(47)=1
f(48)=179
f(49)=101
f(50)=1
f(51)=1
f(52)=1
f(53)=1
f(54)=2939
f(55)=127
f(56)=1
f(57)=409
f(58)=1129
f(59)=1
f(60)=3623
f(61)=1
f(62)=1289
f(63)=499
f(64)=1373
f(65)=1
f(66)=151
f(67)=1
f(68)=1549
f(69)=1
f(70)=547
f(71)=211
f(72)=1
f(73)=223
f(74)=1
f(75)=353
f(76)=1933
f(77)=1
f(78)=197
f(79)=1
f(80)=2141
f(81)=823
f(82)=173
f(83)=1
f(84)=7079
f(85)=1
f(86)=2473
f(87)=1
f(88)=863
f(89)=331
f(90)=8123
f(91)=1
f(92)=1
f(93)=271
f(94)=2953
f(95)=1
f(96)=9239
f(97)=1
f(98)=3209
f(99)=307

b) Substitution of the polynom
The polynom f(x)=x^2+23  could be written as f(y)= y^2+23 with x=y+0

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+0
f'(x)>2x-1  with x > 5

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 : 23, 3, 1, 1, 13, 1, 59, 1, 29, 1, 41, 1, 167, 1, 73, 31, 1, 1, 347, 1,
Found in Database : 23, 3, 13, 59, 29, 41, 167, 73, 31, 347, 47, 599, 233, 269, 71, 349, 139, 131, 1319, 163, 193,
Found in Database : 3, 13, 23, 29, 31, 41, 47, 59, 71, 73, 101, 127, 131, 139,