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Development of Algorithmic Constructions

09:36:18 20.Apr 2024

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Polynom = x^2+7

1. Algorithm

a:=1;
b:=0;
c:=7;


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+7

f(0)=7
f(1)=1
f(2)=11
f(3)=1
f(4)=23
f(5)=1
f(6)=43
f(7)=1
f(8)=71
f(9)=1
f(10)=107
f(11)=1
f(12)=151
f(13)=1
f(14)=29
f(15)=1
f(16)=263
f(17)=37
f(18)=331
f(19)=1
f(20)=1
f(21)=1
f(22)=491
f(23)=67
f(24)=53
f(25)=79
f(26)=683
f(27)=1
f(28)=113
f(29)=1
f(30)=907
f(31)=1
f(32)=1031
f(33)=137
f(34)=1163
f(35)=1
f(36)=1303
f(37)=1
f(38)=1451
f(39)=191
f(40)=1607
f(41)=211
f(42)=1
f(43)=1
f(44)=1
f(45)=127
f(46)=193
f(47)=277
f(48)=2311
f(49)=1
f(50)=109
f(51)=163
f(52)=2711
f(53)=1
f(54)=1
f(55)=379
f(56)=449
f(57)=1
f(58)=3371
f(59)=1
f(60)=3607
f(61)=233
f(62)=3851
f(63)=1
f(64)=373
f(65)=1
f(66)=4363
f(67)=281
f(68)=421
f(69)=149
f(70)=701
f(71)=631
f(72)=179
f(73)=1
f(74)=5483
f(75)=1
f(76)=5783
f(77)=1
f(78)=6091
f(79)=1
f(80)=1
f(81)=821
f(82)=1
f(83)=431
f(84)=1009
f(85)=1
f(86)=673
f(87)=947
f(88)=337
f(89)=991
f(90)=1
f(91)=1
f(92)=197
f(93)=541
f(94)=239
f(95)=1129
f(96)=401
f(97)=1
f(98)=1373
f(99)=613

b) Substitution of the polynom
The polynom f(x)=x^2+7  could be written as f(y)= y^2+7 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 > 3

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 : 7, 1, 11, 1, 23, 1, 43, 1, 71, 1, 107, 1, 151, 1, 29, 1, 263, 37, 331, 1,
Found in Database : 7, 11, 23, 43, 71, 107, 151, 29, 263, 37, 331, 491, 67, 53, 79, 683, 113, 907, 1031, 137, 1163, 1303, 1451, 191,
Found in Database : 7, 11, 23, 29, 37, 43, 53, 67, 71, 79, 107, 109, 113, 127, 137, 149,