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Development of Algorithmic Constructions

10:19:02 19.Apr 2024

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Polynom = x^2+43

1. Algorithm

a:=1;
b:=0;
c:=43;


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+43

f(0)=43
f(1)=11
f(2)=47
f(3)=13
f(4)=59
f(5)=17
f(6)=79
f(7)=23
f(8)=107
f(9)=31
f(10)=1
f(11)=41
f(12)=1
f(13)=53
f(14)=239
f(15)=67
f(16)=1
f(17)=83
f(18)=367
f(19)=101
f(20)=443
f(21)=1
f(22)=1
f(23)=1
f(24)=619
f(25)=167
f(26)=719
f(27)=193
f(28)=827
f(29)=1
f(30)=1
f(31)=251
f(32)=97
f(33)=283
f(34)=109
f(35)=317
f(36)=103
f(37)=353
f(38)=1487
f(39)=1
f(40)=1
f(41)=431
f(42)=139
f(43)=1
f(44)=1979
f(45)=1
f(46)=127
f(47)=563
f(48)=2347
f(49)=1
f(50)=2543
f(51)=661
f(52)=1
f(53)=1
f(54)=269
f(55)=1
f(56)=1
f(57)=823
f(58)=3407
f(59)=881
f(60)=3643
f(61)=941
f(62)=1
f(63)=1
f(64)=4139
f(65)=1
f(66)=1
f(67)=1
f(68)=359
f(69)=1201
f(70)=4943
f(71)=1
f(72)=5227
f(73)=1
f(74)=5519
f(75)=1
f(76)=1
f(77)=1493
f(78)=557
f(79)=1571
f(80)=379
f(81)=1
f(82)=1
f(83)=1733
f(84)=229
f(85)=1
f(86)=173
f(87)=1
f(88)=599
f(89)=181
f(90)=479
f(91)=2081
f(92)=1
f(93)=1
f(94)=683
f(95)=2267
f(96)=197
f(97)=1
f(98)=877
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+43  could be written as f(y)= y^2+43 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 > 7

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 : 43, 11, 47, 13, 59, 17, 79, 23, 107, 31, 1, 41, 1, 53, 239, 67, 1, 83, 367, 101,
Found in Database : 43, 11, 47, 13, 59, 17, 79, 23, 107, 31, 41, 53, 239, 67, 83, 367, 101, 443, 619, 167, 719, 193, 827, 251, 97, 283, 109, 317, 103, 353, 1487,
Found in Database : 11, 13, 17, 23, 31, 41, 43, 47, 53, 59, 67, 79, 83, 97, 101, 103, 107, 109, 127, 139,