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Development of Algorithmic Constructions

20:23:12 18.Apr 2024

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Polynom = x^2+32x+5

1. Algorithm

a:=1;
b:=32;
c:=5;


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+32x+5

f(0)=5
f(1)=19
f(2)=73
f(3)=11
f(4)=149
f(5)=1
f(6)=233
f(7)=139
f(8)=13
f(9)=17
f(10)=1
f(11)=239
f(12)=41
f(13)=59
f(14)=1
f(15)=71
f(16)=773
f(17)=419
f(18)=181
f(19)=487
f(20)=1
f(21)=43
f(22)=1193
f(23)=127
f(24)=1
f(25)=1
f(26)=89
f(27)=47
f(28)=337
f(29)=887
f(30)=373
f(31)=1
f(32)=2053
f(33)=1
f(34)=173
f(35)=1
f(36)=223
f(37)=1279
f(38)=1
f(39)=1
f(40)=577
f(41)=1499
f(42)=283
f(43)=1
f(44)=197
f(45)=347
f(46)=3593
f(47)=1
f(48)=769
f(49)=1987
f(50)=821
f(51)=163
f(52)=4373
f(53)=1
f(54)=4649
f(55)=479
f(56)=4933
f(57)=2539
f(58)=1
f(59)=2687
f(60)=1
f(61)=167
f(62)=307
f(63)=599
f(64)=1
f(65)=631
f(66)=6473
f(67)=3319
f(68)=1361
f(69)=317
f(70)=1429
f(71)=3659
f(72)=1
f(73)=1
f(74)=1
f(75)=1
f(76)=191
f(77)=1
f(78)=101
f(79)=107
f(80)=1
f(81)=241
f(82)=199
f(83)=1
f(84)=9749
f(85)=1
f(86)=1
f(87)=5179
f(88)=2113
f(89)=5387
f(90)=1
f(91)=509
f(92)=113
f(93)=1163
f(94)=1
f(95)=1
f(96)=647
f(97)=569
f(98)=2549
f(99)=499

b) Substitution of the polynom
The polynom f(x)=x^2+32x+5  could be written as f(y)= y^2-251 with x=y-16

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+16
f'(x)>2x+31

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 : 5, 19, 73, 11, 149, 1, 233, 139, 13, 17, 1, 239, 41, 59, 1, 71, 773, 419, 181, 487,
Found in Database : 5, 19, 73, 11, 149, 233, 139, 13, 17, 239, 41, 59, 71, 773, 419, 181, 487, 43, 1193, 127, 89, 47, 337, 887, 373, 2053, 173, 223, 1279,
Found in Database : 5, 11, 13, 17, 19, 41, 43, 47, 59, 71, 73, 89, 101, 107, 113, 127, 139, 149,