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Development of Algorithmic Constructions

17:30:38 19.Apr 2024

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Polynom = x^2+66x-73

1. Algorithm

a:=1;
b:=66;
c:=-73;


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+66x-73

f(0)=73
f(1)=3
f(2)=7
f(3)=67
f(4)=23
f(5)=47
f(6)=359
f(7)=1
f(8)=173
f(9)=43
f(10)=229
f(11)=1
f(12)=863
f(13)=53
f(14)=349
f(15)=571
f(16)=59
f(17)=223
f(18)=1439
f(19)=257
f(20)=61
f(21)=877
f(22)=1
f(23)=1
f(24)=2087
f(25)=367
f(26)=773
f(27)=1
f(28)=853
f(29)=149
f(30)=401
f(31)=163
f(32)=1021
f(33)=1597
f(34)=1109
f(35)=577
f(36)=1
f(37)=89
f(38)=431
f(39)=2011
f(40)=463
f(41)=719
f(42)=4463
f(43)=769
f(44)=227
f(45)=107
f(46)=1693
f(47)=97
f(48)=5399
f(49)=103
f(50)=83
f(51)=421
f(52)=1
f(53)=1039
f(54)=1
f(55)=1097
f(56)=751
f(57)=3469
f(58)=113
f(59)=1217
f(60)=7487
f(61)=1279
f(62)=2621
f(63)=4027
f(64)=2749
f(65)=1
f(66)=1
f(67)=491
f(68)=131
f(69)=4621
f(70)=1
f(71)=1609
f(72)=1409
f(73)=1
f(74)=127
f(75)=1
f(76)=397
f(77)=1823
f(78)=11159
f(79)=271
f(80)=1
f(81)=1
f(82)=4021
f(83)=683
f(84)=12527
f(85)=709
f(86)=619
f(87)=6619
f(88)=4493
f(89)=2287
f(90)=13967
f(91)=1
f(92)=1607
f(93)=1051
f(94)=1663
f(95)=1
f(96)=673
f(97)=1
f(98)=5333
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+66x-73  could be written as f(y)= y^2-1162 with x=y-33

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+33
f'(x)>2x+65

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 : 73, 3, 7, 67, 23, 47, 359, 1, 173, 43, 229, 1, 863, 53, 349, 571, 59, 223, 1439, 257,
Found in Database : 73, 3, 7, 67, 23, 47, 359, 173, 43, 229, 863, 53, 349, 571, 59, 223, 1439, 257, 61, 877, 2087, 367, 773, 853, 149, 401, 163, 1021, 1597, 1109, 577, 89, 431, 2011,
Found in Database : 3, 7, 23, 43, 47, 53, 59, 61, 67, 73, 83, 89, 97, 103, 107, 113, 127, 131, 149,