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Development of Algorithmic Constructions

09:21:58 20.Apr 2024

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Polynom = x^2-152x+1559

1. Algorithm

a:=1;
b:=-152;
c:=1559;


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^2-152x+1559

f(0)=1559
f(1)=11
f(2)=1259
f(3)=139
f(4)=967
f(5)=103
f(6)=683
f(7)=17
f(8)=37
f(9)=1
f(10)=1
f(11)=1
f(12)=1
f(13)=31
f(14)=373
f(15)=1
f(16)=617
f(17)=23
f(18)=853
f(19)=1
f(20)=47
f(21)=149
f(22)=1301
f(23)=1
f(24)=89
f(25)=101
f(26)=1
f(27)=227
f(28)=1913
f(29)=251
f(30)=191
f(31)=137
f(32)=2281
f(33)=1
f(34)=223
f(35)=317
f(36)=2617
f(37)=337
f(38)=59
f(39)=1
f(40)=127
f(41)=1
f(42)=3061
f(43)=1
f(44)=1
f(45)=1
f(46)=107
f(47)=211
f(48)=3433
f(49)=109
f(50)=3541
f(51)=449
f(52)=331
f(53)=461
f(54)=3733
f(55)=1
f(56)=347
f(57)=241
f(58)=229
f(59)=491
f(60)=233
f(61)=499
f(62)=4021
f(63)=1
f(64)=4073
f(65)=1
f(66)=179
f(67)=1
f(68)=4153
f(69)=521
f(70)=113
f(71)=131
f(72)=4201
f(73)=263
f(74)=383
f(75)=1
f(76)=4217
f(77)=1
f(78)=1
f(79)=1
f(80)=1
f(81)=1
f(82)=1
f(83)=1
f(84)=1
f(85)=1
f(86)=1
f(87)=1
f(88)=1
f(89)=1
f(90)=1
f(91)=1
f(92)=1
f(93)=1
f(94)=1
f(95)=1
f(96)=1
f(97)=1
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-152x+1559  could be written as f(y)= y^2-4217 with x=y+76

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-76
f'(x)>2x-153  with x > 65

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 : 1559, 11, 1259, 139, 967, 103, 683, 17, 37, 1, 1, 1, 1, 31, 373, 1, 617, 23, 853, 1,
Found in Database : 1559, 11, 1259, 139, 967, 103, 683, 17, 37, 31, 373, 617, 23, 853, 47, 149, 1301, 89, 101, 227, 1913, 251, 191, 137, 2281, 223, 317, 2617, 337, 59,
Found in Database : 11, 17, 23, 31, 37, 47, 59, 89, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149,