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Development of Algorithmic Constructions

11:01:08 19.Apr 2024

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Polynom = x^2+52x-373

1. Algorithm

a:=1;
b:=52;
c:=-373;


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+52x-373

f(0)=373
f(1)=5
f(2)=53
f(3)=13
f(4)=149
f(5)=11
f(6)=1
f(7)=1
f(8)=107
f(9)=1
f(10)=19
f(11)=1
f(12)=79
f(13)=59
f(14)=29
f(15)=1
f(16)=1
f(17)=1
f(18)=887
f(19)=61
f(20)=97
f(21)=1
f(22)=251
f(23)=1
f(24)=1451
f(25)=1
f(26)=331
f(27)=1
f(28)=1867
f(29)=1
f(30)=2087
f(31)=1
f(32)=463
f(33)=1
f(34)=2551
f(35)=167
f(36)=43
f(37)=73
f(38)=277
f(39)=397
f(40)=3307
f(41)=1
f(42)=1
f(43)=1
f(44)=3851
f(45)=499
f(46)=827
f(47)=1
f(48)=233
f(49)=1
f(50)=163
f(51)=1
f(52)=1
f(53)=1
f(54)=5351
f(55)=1
f(56)=227
f(57)=1
f(58)=6007
f(59)=193
f(60)=577
f(61)=1
f(62)=103
f(63)=859
f(64)=641
f(65)=113
f(66)=1483
f(67)=1
f(68)=599
f(69)=997
f(70)=8167
f(71)=1
f(72)=1
f(73)=547
f(74)=8951
f(75)=1
f(76)=1871
f(77)=239
f(78)=9767
f(79)=1
f(80)=1
f(81)=1
f(82)=1
f(83)=677
f(84)=257
f(85)=1409
f(86)=1
f(87)=293
f(88)=919
f(89)=761
f(90)=653
f(91)=1
f(92)=1
f(93)=1
f(94)=1
f(95)=1699
f(96)=2767
f(97)=1
f(98)=14327
f(99)=911

b) Substitution of the polynom
The polynom f(x)=x^2+52x-373  could be written as f(y)= y^2-1049 with x=y-26

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+26
f'(x)>2x+51

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 : 373, 5, 53, 13, 149, 11, 1, 1, 107, 1, 19, 1, 79, 59, 29, 1, 1, 1, 887, 61,
Found in Database : 373, 5, 53, 13, 149, 11, 107, 19, 79, 59, 29, 887, 61, 97, 251, 1451, 331, 1867, 2087, 463, 2551, 167, 43, 73, 277, 397,
Found in Database : 5, 11, 13, 19, 29, 43, 53, 59, 61, 73, 79, 97, 103, 107, 113, 131, 149,