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Development of Algorithmic Constructions

22:20:26 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)=17
f(2)=5
f(3)=1
f(4)=11
f(5)=29
f(6)=47
f(7)=1
f(8)=193
f(9)=1
f(10)=743
f(11)=23
f(12)=61
f(13)=251
f(14)=73
f(15)=1
f(16)=1283
f(17)=1
f(18)=1
f(19)=79
f(20)=1
f(21)=149
f(22)=379
f(23)=167
f(24)=1
f(25)=557
f(26)=71
f(27)=41
f(28)=2579
f(29)=1
f(30)=941
f(31)=67
f(32)=1
f(33)=89
f(34)=1
f(35)=1
f(36)=1201
f(37)=1
f(38)=431
f(39)=1
f(40)=181
f(41)=359
f(42)=1
f(43)=1151
f(44)=317
f(45)=409
f(46)=83
f(47)=1
f(48)=163
f(49)=277
f(50)=1901
f(51)=1
f(52)=1
f(53)=1
f(54)=1
f(55)=1637
f(56)=1
f(57)=1
f(58)=7079
f(59)=1
f(60)=827
f(61)=1907
f(62)=521
f(63)=1
f(64)=1
f(65)=233
f(66)=2861
f(67)=439
f(68)=1
f(69)=1
f(70)=853
f(71)=1
f(72)=653
f(73)=1
f(74)=227
f(75)=1
f(76)=367
f(77)=1
f(78)=1231
f(79)=113
f(80)=1
f(81)=1
f(82)=479
f(83)=1
f(84)=829
f(85)=3167
f(86)=1
f(87)=1
f(88)=787
f(89)=1
f(90)=4621
f(91)=3527
f(92)=1
f(93)=1217
f(94)=2971
f(95)=1259
f(96)=569
f(97)=1
f(98)=1
f(99)=269

b) Substitution of the polynom
The polynom f(x)=x^2+64x+3  could be written as f(y)= y^2-1021 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, 17, 5, 1, 11, 29, 47, 1, 193, 1, 743, 23, 61, 251, 73, 1, 1283, 1, 1, 79,
Found in Database : 3, 17, 5, 11, 29, 47, 193, 743, 23, 61, 251, 73, 1283, 79, 149, 379, 167, 557, 71, 41, 2579, 941, 67, 89, 1201, 431,
Found in Database : 3, 5, 11, 17, 23, 29, 41, 47, 61, 67, 71, 73, 79, 83, 89, 113, 149,