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Development of Algorithmic Constructions

17:54:42 20.Apr 2024

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Polynom = x^2+16x-3

1. Algorithm

a:=1;
b:=16;
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+16x-3

f(0)=3
f(1)=7
f(2)=11
f(3)=1
f(4)=1
f(5)=17
f(6)=43
f(7)=79
f(8)=1
f(9)=37
f(10)=257
f(11)=1
f(12)=1
f(13)=1
f(14)=139
f(15)=1
f(16)=509
f(17)=31
f(18)=29
f(19)=331
f(20)=239
f(21)=1
f(22)=1
f(23)=149
f(24)=1
f(25)=73
f(26)=1
f(27)=193
f(28)=1229
f(29)=1
f(30)=1
f(31)=727
f(32)=1
f(33)=269
f(34)=1697
f(35)=1
f(36)=89
f(37)=1
f(38)=683
f(39)=1
f(40)=2237
f(41)=389
f(42)=811
f(43)=181
f(44)=293
f(45)=457
f(46)=1
f(47)=1
f(48)=1
f(49)=1
f(50)=157
f(51)=569
f(52)=3533
f(53)=1
f(54)=1259
f(55)=1951
f(56)=1
f(57)=1
f(58)=4289
f(59)=67
f(60)=1
f(61)=2347
f(62)=179
f(63)=829
f(64)=1
f(65)=877
f(66)=601
f(67)=397
f(68)=173
f(69)=977
f(70)=547
f(71)=1
f(72)=2111
f(73)=191
f(74)=317
f(75)=379
f(76)=241
f(77)=1193
f(78)=349
f(79)=1
f(80)=853
f(81)=1
f(82)=277
f(83)=1
f(84)=311
f(85)=613
f(86)=1
f(87)=1493
f(88)=1307
f(89)=1
f(90)=1
f(91)=1
f(92)=1
f(93)=563
f(94)=10337
f(95)=251
f(96)=3583
f(97)=5479
f(98)=1
f(99)=271

b) Substitution of the polynom
The polynom f(x)=x^2+16x-3  could be written as f(y)= y^2-67 with x=y-8

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+8
f'(x)>2x+15

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, 7, 11, 1, 1, 17, 43, 79, 1, 37, 257, 1, 1, 1, 139, 1, 509, 31, 29, 331,
Found in Database : 3, 7, 11, 17, 43, 79, 37, 257, 139, 509, 31, 29, 331, 239, 149, 73, 193, 1229, 727, 269, 1697, 89, 683,
Found in Database : 3, 7, 11, 17, 29, 31, 37, 43, 67, 73, 79, 89, 139, 149,