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Development of Algorithmic Constructions

15:53:22 19.Apr 2024

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Polynom = x^2-180x-53

1. Algorithm

a:=1;
b:=-180;
c:=-53;


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^2-180x-53

f(0)=53
f(1)=29
f(2)=409
f(3)=73
f(4)=757
f(5)=1
f(6)=1097
f(7)=79
f(8)=1429
f(9)=199
f(10)=1753
f(11)=239
f(12)=2069
f(13)=139
f(14)=2377
f(15)=1
f(16)=2677
f(17)=353
f(18)=2969
f(19)=389
f(20)=3253
f(21)=1
f(22)=3529
f(23)=229
f(24)=3797
f(25)=491
f(26)=4057
f(27)=523
f(28)=31
f(29)=277
f(30)=157
f(31)=1
f(32)=4789
f(33)=613
f(34)=173
f(35)=641
f(36)=5237
f(37)=167
f(38)=5449
f(39)=347
f(40)=5653
f(41)=719
f(42)=5849
f(43)=743
f(44)=6037
f(45)=383
f(46)=6217
f(47)=197
f(48)=6389
f(49)=809
f(50)=6553
f(51)=829
f(52)=6709
f(53)=1
f(54)=6857
f(55)=433
f(56)=6997
f(57)=883
f(58)=7129
f(59)=1
f(60)=7253
f(61)=457
f(62)=7369
f(63)=1
f(64)=7477
f(65)=941
f(66)=7577
f(67)=953
f(68)=7669
f(69)=241
f(70)=7753
f(71)=487
f(72)=7829
f(73)=983
f(74)=149
f(75)=991
f(76)=109
f(77)=499
f(78)=8009
f(79)=251
f(80)=8053
f(81)=1009
f(82)=8089
f(83)=1013
f(84)=8117
f(85)=127
f(86)=103
f(87)=509
f(88)=281
f(89)=1019
f(90)=263
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^2-180x-53  could be written as f(y)= y^2-8153 with x=y+90

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-90
f'(x)>2x-181  with x > 90

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 : 53, 29, 409, 73, 757, 1, 1097, 79, 1429, 199, 1753, 239, 2069, 139, 2377, 1, 2677, 353, 2969, 389,
Found in Database : 53, 29, 409, 73, 757, 1097, 79, 1429, 199, 1753, 239, 2069, 139, 2377, 2677, 353, 2969, 389, 3253, 3529, 229, 3797, 491, 4057, 523, 31, 277, 157, 4789, 613, 173, 641, 5237, 167, 5449, 347,
Found in Database : 29, 31, 53, 73, 79, 103, 109, 127, 139, 149,