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Development of Algorithmic Constructions

00:29:08 20.Apr 2024

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Polynom = x^2-230x-3

1. Algorithm

a:=1;
b:=-230;
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^2-230x-3

f(0)=3
f(1)=29
f(2)=17
f(3)=19
f(4)=907
f(5)=47
f(6)=449
f(7)=23
f(8)=593
f(9)=83
f(10)=2203
f(11)=67
f(12)=97
f(13)=353
f(14)=1009
f(15)=269
f(16)=149
f(17)=151
f(18)=1
f(19)=59
f(20)=467
f(21)=61
f(22)=241
f(23)=397
f(24)=1
f(25)=641
f(26)=1
f(27)=457
f(28)=5659
f(29)=1
f(30)=1
f(31)=1543
f(32)=2113
f(33)=271
f(34)=113
f(35)=569
f(36)=137
f(37)=1
f(38)=811
f(39)=1
f(40)=7603
f(41)=1
f(42)=2633
f(43)=2011
f(44)=2729
f(45)=347
f(46)=8467
f(47)=239
f(48)=971
f(49)=1109
f(50)=3001
f(51)=761
f(52)=197
f(53)=1
f(54)=3169
f(55)=1
f(56)=1
f(57)=1
f(58)=587
f(59)=1
f(60)=179
f(61)=1289
f(62)=1
f(63)=877
f(64)=10627
f(65)=1
f(66)=401
f(67)=2731
f(68)=3673
f(69)=463
f(70)=659
f(71)=941
f(72)=3793
f(73)=1433
f(74)=1283
f(75)=1
f(76)=509
f(77)=491
f(78)=1
f(79)=157
f(80)=4001
f(81)=503
f(82)=199
f(83)=1
f(84)=1
f(85)=1
f(86)=4129
f(87)=1
f(88)=431
f(89)=523
f(90)=4201
f(91)=3163
f(92)=1
f(93)=1
f(94)=673
f(95)=1069
f(96)=4289
f(97)=1613
f(98)=227
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-230x-3  could be written as f(y)= y^2-13228 with x=y+115

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-115
f'(x)>2x-231  with x > 115

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, 29, 17, 19, 907, 47, 449, 23, 593, 83, 2203, 67, 97, 353, 1009, 269, 149, 151, 1, 59,
Found in Database : 3, 29, 17, 19, 907, 47, 449, 23, 593, 83, 2203, 67, 97, 353, 1009, 269, 149, 151, 59, 467, 61, 241, 397, 641, 457, 5659, 1543, 2113, 271, 113, 569, 137, 811,
Found in Database : 3, 17, 19, 23, 29, 47, 59, 61, 67, 83, 97, 113, 137, 149,