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Development of Algorithmic Constructions

19:29:44 19.Apr 2024

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Polynom = x^2-268x+3083

1. Algorithm

a:=1;
b:=-268;
c:=3083;


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^2-268x+3083

f(0)=3083
f(1)=11
f(2)=2551
f(3)=13
f(4)=2027
f(5)=17
f(6)=1511
f(7)=157
f(8)=59
f(9)=47
f(10)=503
f(11)=1
f(12)=1
f(13)=29
f(14)=43
f(15)=89
f(16)=73
f(17)=37
f(18)=109
f(19)=103
f(20)=1877
f(21)=263
f(22)=137
f(23)=1
f(24)=1
f(25)=1
f(26)=3209
f(27)=107
f(28)=3637
f(29)=1
f(30)=4057
f(31)=41
f(32)=1
f(33)=1
f(34)=443
f(35)=317
f(36)=479
f(37)=683
f(38)=5657
f(39)=1
f(40)=6037
f(41)=389
f(42)=1
f(43)=1
f(44)=521
f(45)=79
f(46)=7129
f(47)=83
f(48)=7477
f(49)=239
f(50)=7817
f(51)=499
f(52)=281
f(53)=1039
f(54)=229
f(55)=1
f(56)=1
f(57)=1
f(58)=827
f(59)=1
f(60)=9397
f(61)=1193
f(62)=9689
f(63)=1229
f(64)=9973
f(65)=1
f(66)=277
f(67)=1
f(68)=809
f(69)=1
f(70)=829
f(71)=1
f(72)=269
f(73)=1
f(74)=11273
f(75)=1
f(76)=677
f(77)=1453
f(78)=97
f(79)=1481
f(80)=1087
f(81)=1
f(82)=283
f(83)=1
f(84)=12373
f(85)=1559
f(86)=12569
f(87)=1583
f(88)=12757
f(89)=1
f(90)=761
f(91)=1
f(92)=13109
f(93)=1
f(94)=1021
f(95)=1669
f(96)=1033
f(97)=211
f(98)=13577
f(99)=853

b) Substitution of the polynom
The polynom f(x)=x^2-268x+3083  could be written as f(y)= y^2-14873 with x=y+134

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-134
f'(x)>2x-269  with x > 122

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 : 3083, 11, 2551, 13, 2027, 17, 1511, 157, 59, 47, 503, 1, 1, 29, 43, 89, 73, 37, 109, 103,
Found in Database : 3083, 11, 2551, 13, 2027, 17, 1511, 157, 59, 47, 503, 29, 43, 89, 73, 37, 109, 103, 1877, 263, 137, 3209, 107, 3637, 4057, 41, 443, 317, 479, 683, 5657,
Found in Database : 11, 13, 17, 29, 37, 41, 43, 47, 59, 73, 79, 83, 89, 97, 103, 107, 109, 137, 139,