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Development of Algorithmic Constructions

10:55:14 20.Apr 2024

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Polynom = x^2+22x-37

1. Algorithm

a:=1;
b:=22;
c:=-37;


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+22x-37

f(0)=37
f(1)=7
f(2)=11
f(3)=19
f(4)=67
f(5)=1
f(6)=131
f(7)=83
f(8)=29
f(9)=1
f(10)=283
f(11)=163
f(12)=53
f(13)=1
f(14)=467
f(15)=1
f(16)=571
f(17)=313
f(18)=683
f(19)=1
f(20)=73
f(21)=433
f(22)=1
f(23)=499
f(24)=97
f(25)=569
f(26)=173
f(27)=643
f(28)=47
f(29)=103
f(30)=1523
f(31)=1
f(32)=89
f(33)=127
f(34)=1867
f(35)=1
f(36)=293
f(37)=1
f(38)=2243
f(39)=1171
f(40)=349
f(41)=1
f(42)=241
f(43)=197
f(44)=61
f(45)=1489
f(46)=281
f(47)=229
f(48)=3323
f(49)=1721
f(50)=509
f(51)=1
f(52)=1
f(53)=179
f(54)=1
f(55)=2099
f(56)=71
f(57)=1
f(58)=4603
f(59)=2371
f(60)=257
f(61)=359
f(62)=5171
f(63)=2659
f(64)=1
f(65)=1
f(66)=199
f(67)=2963
f(68)=79
f(69)=3121
f(70)=337
f(71)=1
f(72)=1
f(73)=3449
f(74)=191
f(75)=1
f(76)=7411
f(77)=3793
f(78)=1109
f(79)=1
f(80)=8123
f(81)=4153
f(82)=1213
f(83)=4339
f(84)=8867
f(85)=647
f(86)=1
f(87)=4723
f(88)=9643
f(89)=1
f(90)=1
f(91)=109
f(92)=1493
f(93)=1
f(94)=10867
f(95)=1
f(96)=1613
f(97)=523
f(98)=617
f(99)=853

b) Substitution of the polynom
The polynom f(x)=x^2+22x-37  could be written as f(y)= y^2-158 with x=y-11

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+11
f'(x)>2x+21

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 : 37, 7, 11, 19, 67, 1, 131, 83, 29, 1, 283, 163, 53, 1, 467, 1, 571, 313, 683, 1,
Found in Database : 37, 7, 11, 19, 67, 131, 83, 29, 283, 163, 53, 467, 571, 313, 683, 73, 433, 499, 97, 569, 173, 643, 47, 103, 1523, 89, 127, 1867, 293, 2243, 1171,
Found in Database : 7, 11, 19, 29, 37, 47, 53, 61, 67, 71, 73, 79, 83, 89, 97, 103, 109, 127, 131, 149,