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Development of Algorithmic Constructions

14:03:30 20.Apr 2024

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Polynom = x^2+72x+23

1. Algorithm

a:=1;
b:=72;
c:=23;


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+72x+23

f(0)=23
f(1)=3
f(2)=19
f(3)=31
f(4)=109
f(5)=17
f(6)=491
f(7)=1
f(8)=13
f(9)=47
f(10)=281
f(11)=1
f(12)=1031
f(13)=1
f(14)=409
f(15)=83
f(16)=53
f(17)=1
f(18)=1
f(19)=73
f(20)=1
f(21)=1
f(22)=41
f(23)=1
f(24)=179
f(25)=1
f(26)=857
f(27)=337
f(28)=941
f(29)=1
f(30)=3083
f(31)=67
f(32)=1117
f(33)=1
f(34)=1
f(35)=157
f(36)=3911
f(37)=1
f(38)=467
f(39)=1
f(40)=79
f(41)=97
f(42)=283
f(43)=1
f(44)=1709
f(45)=661
f(46)=1
f(47)=1
f(48)=5783
f(49)=1
f(50)=1
f(51)=787
f(52)=719
f(53)=277
f(54)=6827
f(55)=1
f(56)=1
f(57)=461
f(58)=2521
f(59)=1
f(60)=1
f(61)=113
f(62)=2777
f(63)=1
f(64)=2909
f(65)=1
f(66)=397
f(67)=389
f(68)=3181
f(69)=1
f(70)=1
f(71)=1
f(72)=10391
f(73)=1
f(74)=401
f(75)=1381
f(76)=1
f(77)=479
f(78)=617
f(79)=1
f(80)=131
f(81)=1
f(82)=4217
f(83)=1
f(84)=13127
f(85)=557
f(86)=349
f(87)=433
f(88)=1567
f(89)=1
f(90)=859
f(91)=619
f(92)=1
f(93)=1
f(94)=5209
f(95)=331
f(96)=521
f(97)=1
f(98)=1
f(99)=163

b) Substitution of the polynom
The polynom f(x)=x^2+72x+23  could be written as f(y)= y^2-1273 with x=y-36

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+36
f'(x)>2x+71

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 : 23, 3, 19, 31, 109, 17, 491, 1, 13, 47, 281, 1, 1031, 1, 409, 83, 53, 1, 1, 73,
Found in Database : 23, 3, 19, 31, 109, 17, 491, 13, 47, 281, 1031, 409, 83, 53, 73, 41, 179, 857, 337, 941, 3083, 67, 1117, 157, 3911, 467,
Found in Database : 3, 13, 17, 19, 23, 31, 41, 47, 53, 67, 73, 79, 83, 97, 109, 113, 131, 149,