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Development of Algorithmic Constructions

12:27:55 20.Apr 2024

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Polynom = x^2-180x+787

1. Algorithm

a:=1;
b:=-180;
c:=787;


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+787

f(0)=787
f(1)=19
f(2)=431
f(3)=1
f(4)=83
f(5)=11
f(6)=257
f(7)=53
f(8)=31
f(9)=47
f(10)=1
f(11)=67
f(12)=1229
f(13)=173
f(14)=29
f(15)=211
f(16)=167
f(17)=1
f(18)=2129
f(19)=71
f(20)=127
f(21)=1
f(22)=2689
f(23)=353
f(24)=2957
f(25)=193
f(26)=3217
f(27)=1
f(28)=3469
f(29)=449
f(30)=79
f(31)=479
f(32)=359
f(33)=1
f(34)=4177
f(35)=1
f(36)=4397
f(37)=563
f(38)=419
f(39)=1
f(40)=4813
f(41)=307
f(42)=5009
f(43)=1
f(44)=5197
f(45)=661
f(46)=283
f(47)=683
f(48)=179
f(49)=1
f(50)=197
f(51)=181
f(52)=5869
f(53)=743
f(54)=547
f(55)=761
f(56)=131
f(57)=389
f(58)=331
f(59)=397
f(60)=1
f(61)=809
f(62)=6529
f(63)=823
f(64)=6637
f(65)=1
f(66)=6737
f(67)=1
f(68)=6829
f(69)=859
f(70)=223
f(71)=1
f(72)=241
f(73)=439
f(74)=7057
f(75)=443
f(76)=647
f(77)=1
f(78)=107
f(79)=1
f(80)=7213
f(81)=113
f(82)=659
f(83)=227
f(84)=383
f(85)=911
f(86)=7297
f(87)=1
f(88)=7309
f(89)=457
f(90)=103
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+787  could be written as f(y)= y^2-7313 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 > 86

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 : 787, 19, 431, 1, 83, 11, 257, 53, 31, 47, 1, 67, 1229, 173, 29, 211, 167, 1, 2129, 71,
Found in Database : 787, 19, 431, 83, 11, 257, 53, 31, 47, 67, 1229, 173, 29, 211, 167, 2129, 71, 127, 2689, 353, 2957, 193, 3217, 3469, 449, 79, 479, 359, 4177, 4397, 563, 419,
Found in Database : 11, 19, 29, 31, 47, 53, 67, 71, 79, 83, 103, 107, 113, 127, 131,