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Development of Algorithmic Constructions

00:21:37 19.Apr 2024

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Polynom = x^2+64x-3

1. Algorithm

a:=1;
b:=64;
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+64x-3

f(0)=3
f(1)=31
f(2)=43
f(3)=11
f(4)=269
f(5)=19
f(6)=139
f(7)=13
f(8)=191
f(9)=109
f(10)=67
f(11)=137
f(12)=101
f(13)=499
f(14)=1
f(15)=197
f(16)=1277
f(17)=229
f(18)=491
f(19)=787
f(20)=1
f(21)=1
f(22)=1889
f(23)=37
f(24)=1
f(25)=1
f(26)=41
f(27)=409
f(28)=83
f(29)=449
f(30)=313
f(31)=1471
f(32)=1
f(33)=1
f(34)=3329
f(35)=577
f(36)=1
f(37)=1867
f(38)=1291
f(39)=223
f(40)=4157
f(41)=239
f(42)=1483
f(43)=1
f(44)=1583
f(45)=1
f(46)=389
f(47)=79
f(48)=199
f(49)=2767
f(50)=211
f(51)=977
f(52)=6029
f(53)=1033
f(54)=193
f(55)=3271
f(56)=2239
f(57)=383
f(58)=643
f(59)=1
f(60)=1
f(61)=103
f(62)=1
f(63)=1
f(64)=431
f(65)=127
f(66)=953
f(67)=107
f(68)=997
f(69)=1
f(70)=9377
f(71)=1597
f(72)=251
f(73)=4999
f(74)=1
f(75)=1
f(76)=967
f(77)=1
f(78)=3691
f(79)=5647
f(80)=349
f(81)=1
f(82)=11969
f(83)=1
f(84)=1381
f(85)=487
f(86)=1433
f(87)=1
f(88)=311
f(89)=2269
f(90)=149
f(91)=641
f(92)=4783
f(93)=811
f(94)=479
f(95)=839
f(96)=5119
f(97)=1
f(98)=1
f(99)=2689

b) Substitution of the polynom
The polynom f(x)=x^2+64x-3  could be written as f(y)= y^2-1027 with x=y-32

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+32
f'(x)>2x+63

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, 31, 43, 11, 269, 19, 139, 13, 191, 109, 67, 137, 101, 499, 1, 197, 1277, 229, 491, 787,
Found in Database : 3, 31, 43, 11, 269, 19, 139, 13, 191, 109, 67, 137, 101, 499, 197, 1277, 229, 491, 787, 1889, 37, 41, 409, 83, 449, 313, 1471, 3329, 577, 1867, 1291, 223,
Found in Database : 3, 11, 13, 19, 31, 37, 41, 43, 67, 79, 83, 101, 103, 107, 109, 127, 137, 139, 149,