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Development of Algorithmic Constructions

05:21:18 20.Apr 2024

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Polynom = x^2+120x-449

1. Algorithm

a:=1;
b:=120;
c:=-449;


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+120x-449

f(0)=449
f(1)=41
f(2)=5
f(3)=1
f(4)=47
f(5)=11
f(6)=307
f(7)=1
f(8)=23
f(9)=89
f(10)=37
f(11)=31
f(12)=227
f(13)=1
f(14)=1427
f(15)=197
f(16)=157
f(17)=1
f(18)=1
f(19)=137
f(20)=2351
f(21)=1
f(22)=107
f(23)=71
f(24)=97
f(25)=397
f(26)=3347
f(27)=1
f(28)=739
f(29)=1
f(30)=4051
f(31)=1
f(32)=883
f(33)=1
f(34)=4787
f(35)=311
f(36)=5167
f(37)=67
f(38)=101
f(39)=719
f(40)=541
f(41)=769
f(42)=1
f(43)=1
f(44)=1
f(45)=109
f(46)=7187
f(47)=1
f(48)=1523
f(49)=1
f(50)=83
f(51)=1
f(52)=1699
f(53)=1
f(54)=389
f(55)=1
f(56)=409
f(57)=241
f(58)=79
f(59)=1
f(60)=941
f(61)=331
f(62)=1
f(63)=277
f(64)=1
f(65)=1447
f(66)=11827
f(67)=151
f(68)=2467
f(69)=787
f(70)=181
f(71)=149
f(72)=1
f(73)=1
f(74)=13907
f(75)=443
f(76)=14447
f(77)=1
f(78)=2999
f(79)=1
f(80)=15551
f(81)=1979
f(82)=293
f(83)=1
f(84)=1
f(85)=1061
f(86)=557
f(87)=439
f(88)=3571
f(89)=2269
f(90)=18451
f(91)=1
f(92)=103
f(93)=1
f(94)=1
f(95)=1
f(96)=20287
f(97)=1
f(98)=1
f(99)=1327

b) Substitution of the polynom
The polynom f(x)=x^2+120x-449  could be written as f(y)= y^2-4049 with x=y-60

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+60
f'(x)>2x+119

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 : 449, 41, 5, 1, 47, 11, 307, 1, 23, 89, 37, 31, 227, 1, 1427, 197, 157, 1, 1, 137,
Found in Database : 449, 41, 5, 47, 11, 307, 23, 89, 37, 31, 227, 1427, 197, 157, 137, 2351, 107, 71, 97, 397, 3347, 739, 4051, 883, 4787, 311, 5167, 67, 101, 719,
Found in Database : 5, 11, 23, 31, 37, 41, 47, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 137, 149,