---

Development of Algorithmic Constructions

15:34:37 20.Apr 2024

---

Polynom = x^2+19x-7

1. Algorithm

a:=1;
b:=19;
c:=-7;


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+19x-7

f(0)=7
f(1)=13
f(2)=5
f(3)=59
f(4)=17
f(5)=113
f(6)=11
f(7)=1
f(8)=19
f(9)=1
f(10)=283
f(11)=1
f(12)=73
f(13)=409
f(14)=1
f(15)=503
f(16)=79
f(17)=1
f(18)=659
f(19)=1
f(20)=773
f(21)=1
f(22)=179
f(23)=137
f(24)=41
f(25)=1093
f(26)=1163
f(27)=1
f(28)=1
f(29)=277
f(30)=1
f(31)=1543
f(32)=1
f(33)=1709
f(34)=359
f(35)=269
f(36)=1973
f(37)=1
f(38)=127
f(39)=1
f(40)=181
f(41)=223
f(42)=1
f(43)=2659
f(44)=1
f(45)=1
f(46)=157
f(47)=619
f(48)=3209
f(49)=1
f(50)=313
f(51)=509
f(52)=67
f(53)=293
f(54)=787
f(55)=239
f(56)=599
f(57)=173
f(58)=1
f(59)=919
f(60)=4733
f(61)=443
f(62)=1
f(63)=1
f(64)=1061
f(65)=1
f(66)=431
f(67)=1151
f(68)=311
f(69)=1213
f(70)=1
f(71)=491
f(72)=1
f(73)=6709
f(74)=1
f(75)=7043
f(76)=7213
f(77)=211
f(78)=7559
f(79)=1
f(80)=193
f(81)=8093
f(82)=331
f(83)=769
f(84)=1
f(85)=1
f(86)=1289
f(87)=97
f(88)=1
f(89)=1
f(90)=9803
f(91)=1429
f(92)=1
f(93)=1487
f(94)=1
f(95)=1
f(96)=1
f(97)=1
f(98)=1637
f(99)=467

b) Substitution of the polynom
The polynom f(x)=x^2+19x-7  could be written as f(y)= y^2-97.25 with x=y-9.5

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+9.5
f'(x)>2x+18

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 : 7, 13, 5, 59, 17, 113, 11, 1, 19, 1, 283, 1, 73, 409, 1, 503, 79, 1, 659, 1,
Found in Database : 7, 13, 5, 59, 17, 113, 11, 19, 283, 73, 409, 503, 79, 659, 773, 179, 137, 41, 1093, 1163, 277, 1543, 1709, 359, 269, 1973, 127,
Found in Database : 5, 7, 11, 13, 17, 19, 41, 59, 67, 73, 79, 97, 113, 127, 137,