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Development of Algorithmic Constructions

15:42:15 20.Apr 2024

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Polynom = x^2-37x+3

1. Algorithm

a:=1;
b:=-37;
c:=3;


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^2-37x+3

f(0)=3
f(1)=11
f(2)=67
f(3)=1
f(4)=43
f(5)=157
f(6)=61
f(7)=23
f(8)=229
f(9)=83
f(10)=89
f(11)=283
f(12)=1
f(13)=103
f(14)=29
f(15)=109
f(16)=37
f(17)=337
f(18)=113
f(19)=1
f(20)=1
f(21)=1
f(22)=1
f(23)=1
f(24)=1
f(25)=1
f(26)=1
f(27)=1
f(28)=1
f(29)=1
f(30)=1
f(31)=1
f(32)=1
f(33)=1
f(34)=1
f(35)=1
f(36)=1
f(37)=1
f(38)=41
f(39)=1
f(40)=1
f(41)=167
f(42)=71
f(43)=1
f(44)=311
f(45)=1
f(46)=139
f(47)=1
f(48)=59
f(49)=197
f(50)=653
f(51)=239
f(52)=1
f(53)=1
f(54)=307
f(55)=331
f(56)=97
f(57)=127
f(58)=1
f(59)=1301
f(60)=461
f(61)=163
f(62)=1553
f(63)=547
f(64)=577
f(65)=1823
f(66)=1
f(67)=1
f(68)=2111
f(69)=1
f(70)=257
f(71)=2417
f(72)=1
f(73)=877
f(74)=2741
f(75)=317
f(76)=1
f(77)=3083
f(78)=1
f(79)=1
f(80)=313
f(81)=1
f(82)=1231
f(83)=3821
f(84)=439
f(85)=1361
f(86)=4217
f(87)=1451
f(88)=499
f(89)=421
f(90)=1
f(91)=149
f(92)=1
f(93)=193
f(94)=1787
f(95)=1
f(96)=1889
f(97)=647
f(98)=5981
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-37x+3  could be written as f(y)= y^2-339.25 with x=y+18.5

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-18.5
f'(x)>2x-38  with x > 18

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 : 3, 11, 67, 1, 43, 157, 61, 23, 229, 83, 89, 283, 1, 103, 29, 109, 37, 337, 113, 1,
Found in Database : 3, 11, 67, 43, 157, 61, 23, 229, 83, 89, 283, 103, 29, 109, 37, 337, 113, 41,
Found in Database : 3, 11, 23, 29, 37, 41, 43, 59, 61, 67, 71, 83, 89, 97, 103, 109, 113, 127, 139, 149,