---

Development of Algorithmic Constructions

01:14:19 20.Apr 2024

---

Polynom = x^2+36x-13

1. Algorithm

a:=1;
b:=36;
c:=-13;


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+36x-13

f(0)=13
f(1)=3
f(2)=7
f(3)=1
f(4)=1
f(5)=1
f(6)=239
f(7)=1
f(8)=113
f(9)=1
f(10)=149
f(11)=1
f(12)=563
f(13)=1
f(14)=229
f(15)=47
f(16)=1
f(17)=37
f(18)=137
f(19)=43
f(20)=41
f(21)=1
f(22)=421
f(23)=1
f(24)=1427
f(25)=1
f(26)=1
f(27)=211
f(28)=593
f(29)=1
f(30)=281
f(31)=1
f(32)=103
f(33)=283
f(34)=263
f(35)=1
f(36)=2579
f(37)=1
f(38)=311
f(39)=1
f(40)=1009
f(41)=131
f(42)=251
f(43)=1
f(44)=167
f(45)=227
f(46)=179
f(47)=1
f(48)=4019
f(49)=173
f(50)=1429
f(51)=79
f(52)=1
f(53)=1
f(54)=1
f(55)=1
f(56)=571
f(57)=661
f(58)=1
f(59)=233
f(60)=821
f(61)=1
f(62)=1
f(63)=389
f(64)=2129
f(65)=1
f(66)=6719
f(67)=1
f(68)=181
f(69)=1
f(70)=823
f(71)=1
f(72)=1109
f(73)=331
f(74)=1
f(75)=1039
f(76)=2833
f(77)=1
f(78)=683
f(79)=1
f(80)=3089
f(81)=1
f(82)=3221
f(83)=1
f(84)=10067
f(85)=107
f(86)=499
f(87)=1
f(88)=1
f(89)=463
f(90)=241
f(91)=1
f(92)=1307
f(93)=1
f(94)=313
f(95)=1
f(96)=12659
f(97)=1
f(98)=4373
f(99)=1669

b) Substitution of the polynom
The polynom f(x)=x^2+36x-13  could be written as f(y)= y^2-337 with x=y-18

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+18
f'(x)>2x+35

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 : 13, 3, 7, 1, 1, 1, 239, 1, 113, 1, 149, 1, 563, 1, 229, 47, 1, 37, 137, 43,
Found in Database : 13, 3, 7, 239, 113, 149, 563, 229, 47, 37, 137, 43, 41, 421, 1427, 211, 593, 281, 103, 283, 263, 2579, 311,
Found in Database : 3, 7, 13, 37, 41, 43, 47, 79, 103, 107, 113, 131, 137, 149,