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Development of Algorithmic Constructions

04:32:35 20.Apr 2024

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Polynom = x^2-29x+7

1. Algorithm

a:=1;
b:=-29;
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^2-29x+7

f(0)=7
f(1)=3
f(2)=47
f(3)=71
f(4)=31
f(5)=113
f(6)=131
f(7)=1
f(8)=23
f(9)=173
f(10)=61
f(11)=191
f(12)=197
f(13)=67
f(14)=29
f(15)=1
f(16)=1
f(17)=1
f(18)=1
f(19)=1
f(20)=1
f(21)=1
f(22)=1
f(23)=1
f(24)=1
f(25)=1
f(26)=1
f(27)=1
f(28)=1
f(29)=1
f(30)=37
f(31)=1
f(32)=103
f(33)=139
f(34)=59
f(35)=1
f(36)=1
f(37)=101
f(38)=349
f(39)=397
f(40)=149
f(41)=499
f(42)=79
f(43)=1
f(44)=1
f(45)=727
f(46)=263
f(47)=853
f(48)=919
f(49)=1
f(50)=151
f(51)=1129
f(52)=401
f(53)=1279
f(54)=1
f(55)=479
f(56)=1
f(57)=229
f(58)=563
f(59)=1777
f(60)=1867
f(61)=653
f(62)=2053
f(63)=307
f(64)=107
f(65)=2347
f(66)=1
f(67)=1
f(68)=2659
f(69)=2767
f(70)=137
f(71)=1
f(72)=1
f(73)=1
f(74)=1
f(75)=3457
f(76)=1193
f(77)=1
f(78)=547
f(79)=1319
f(80)=1
f(81)=4219
f(82)=1451
f(83)=1
f(84)=661
f(85)=227
f(86)=4909
f(87)=163
f(88)=1733
f(89)=5347
f(90)=239
f(91)=269
f(92)=829
f(93)=1
f(94)=2039
f(95)=6277
f(96)=1
f(97)=1
f(98)=967
f(99)=991

b) Substitution of the polynom
The polynom f(x)=x^2-29x+7  could be written as f(y)= y^2-203.25 with x=y+14.5

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-14.5
f'(x)>2x-30  with x > 14

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, 3, 47, 71, 31, 113, 131, 1, 23, 173, 61, 191, 197, 67, 29, 1, 1, 1, 1, 1,
Found in Database : 7, 3, 47, 71, 31, 113, 131, 23, 173, 61, 191, 197, 67, 29, 37, 103, 139, 59, 101, 349, 397,
Found in Database : 3, 7, 23, 29, 31, 37, 47, 59, 61, 67, 71, 79, 101, 103, 107, 113, 131, 137, 139, 149,