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Development of Algorithmic Constructions

05:34:15 19.Apr 2024

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Polynom = x^2+58x-3

1. Algorithm

a:=1;
b:=58;
c:=-3;


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+58x-3

f(0)=3
f(1)=7
f(2)=13
f(3)=5
f(4)=1
f(5)=1
f(6)=127
f(7)=113
f(8)=1
f(9)=1
f(10)=677
f(11)=1
f(12)=31
f(13)=23
f(14)=67
f(15)=1
f(16)=1181
f(17)=53
f(18)=1
f(19)=73
f(20)=173
f(21)=1
f(22)=251
f(23)=1
f(24)=131
f(25)=37
f(26)=727
f(27)=191
f(28)=1
f(29)=1
f(30)=293
f(31)=1
f(32)=137
f(33)=1
f(34)=1
f(35)=271
f(36)=1
f(37)=439
f(38)=1
f(39)=1
f(40)=3917
f(41)=1
f(42)=1399
f(43)=1
f(44)=1
f(45)=193
f(46)=683
f(47)=1
f(48)=1
f(49)=1
f(50)=257
f(51)=463
f(52)=5717
f(53)=1
f(54)=1
f(55)=1553
f(56)=709
f(57)=1
f(58)=269
f(59)=1
f(60)=337
f(61)=907
f(62)=1
f(63)=1
f(64)=223
f(65)=1
f(66)=101
f(67)=1
f(68)=571
f(69)=1
f(70)=1
f(71)=109
f(72)=3119
f(73)=239
f(74)=1
f(75)=277
f(76)=10181
f(77)=433
f(78)=1
f(79)=541
f(80)=283
f(81)=1
f(82)=499
f(83)=1
f(84)=1
f(85)=1
f(86)=4127
f(87)=1051
f(88)=367
f(89)=1
f(90)=1
f(91)=3389
f(92)=1
f(93)=1
f(94)=2857
f(95)=1
f(96)=379
f(97)=1879
f(98)=1019
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+58x-3  could be written as f(y)= y^2-844 with x=y-29

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+29
f'(x)>2x+57

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 : 3, 7, 13, 5, 1, 1, 127, 113, 1, 1, 677, 1, 31, 23, 67, 1, 1181, 53, 1, 73,
Found in Database : 3, 7, 13, 5, 127, 113, 677, 31, 23, 67, 1181, 53, 73, 173, 251, 131, 37, 727, 191, 293, 137, 271, 439,
Found in Database : 3, 5, 7, 13, 23, 31, 37, 53, 67, 73, 101, 109, 113, 127, 131, 137,