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Development of Algorithmic Constructions

04:35:54 20.Apr 2024

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Polynom = x^2+79

1. Algorithm

a:=1;
b:=0;
c:=79;


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+79

f(0)=79
f(1)=5
f(2)=83
f(3)=11
f(4)=19
f(5)=13
f(6)=23
f(7)=1
f(8)=1
f(9)=1
f(10)=179
f(11)=1
f(12)=223
f(13)=31
f(14)=1
f(15)=1
f(16)=67
f(17)=1
f(18)=1
f(19)=1
f(20)=479
f(21)=1
f(22)=563
f(23)=1
f(24)=131
f(25)=1
f(26)=151
f(27)=101
f(28)=863
f(29)=1
f(30)=89
f(31)=1
f(32)=1103
f(33)=73
f(34)=1
f(35)=163
f(36)=1
f(37)=181
f(38)=1523
f(39)=1
f(40)=1
f(41)=1
f(42)=97
f(43)=241
f(44)=1
f(45)=263
f(46)=439
f(47)=1
f(48)=2383
f(49)=1
f(50)=2579
f(51)=1
f(52)=1
f(53)=1
f(54)=599
f(55)=1
f(56)=643
f(57)=1
f(58)=313
f(59)=1
f(60)=283
f(61)=1
f(62)=3923
f(63)=1
f(64)=167
f(65)=269
f(66)=887
f(67)=571
f(68)=4703
f(69)=1
f(70)=383
f(71)=1
f(72)=277
f(73)=1
f(74)=1
f(75)=1
f(76)=1171
f(77)=751
f(78)=6163
f(79)=1
f(80)=1
f(81)=1
f(82)=6803
f(83)=1
f(84)=1427
f(85)=1
f(86)=1
f(87)=239
f(88)=7823
f(89)=1
f(90)=8179
f(91)=1
f(92)=8543
f(93)=1091
f(94)=1783
f(95)=569
f(96)=1
f(97)=593
f(98)=421
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+79  could be written as f(y)= y^2+79 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 > 9

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 : 79, 5, 83, 11, 19, 13, 23, 1, 1, 1, 179, 1, 223, 31, 1, 1, 67, 1, 1, 1,
Found in Database : 79, 5, 83, 11, 19, 13, 23, 179, 223, 31, 67, 479, 563, 131, 151, 101, 863, 89, 1103, 73, 163, 181, 1523,
Found in Database : 5, 11, 13, 19, 23, 31, 67, 73, 79, 83, 89, 97, 101, 131,