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Development of Algorithmic Constructions

20:41:43 16.Apr 2024

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Polynom = x^2+176x-1609

1. Algorithm

a:=1;
b:=176;
c:=-1609;


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+176x-1609

f(0)=1609
f(1)=179
f(2)=7
f(3)=67
f(4)=127
f(5)=11
f(6)=47
f(7)=41
f(8)=137
f(9)=1
f(10)=251
f(11)=1
f(12)=647
f(13)=53
f(14)=1051
f(15)=157
f(16)=19
f(17)=1
f(18)=269
f(19)=131
f(20)=2311
f(21)=79
f(22)=1
f(23)=1
f(24)=3191
f(25)=61
f(26)=3643
f(27)=1
f(28)=373
f(29)=271
f(30)=653
f(31)=601
f(32)=103
f(33)=661
f(34)=5531
f(35)=1
f(36)=317
f(37)=1
f(38)=593
f(39)=1
f(40)=89
f(41)=911
f(42)=7547
f(43)=1
f(44)=1153
f(45)=521
f(46)=1229
f(47)=1109
f(48)=223
f(49)=107
f(50)=881
f(51)=1
f(52)=10247
f(53)=1
f(54)=569
f(55)=73
f(56)=11383
f(57)=1459
f(58)=1709
f(59)=383
f(60)=163
f(61)=1
f(62)=13147
f(63)=1
f(64)=13751
f(65)=1
f(66)=1
f(67)=1
f(68)=14983
f(69)=239
f(70)=233
f(71)=181
f(72)=211
f(73)=109
f(74)=1
f(75)=1
f(76)=331
f(77)=1117
f(78)=167
f(79)=1
f(80)=113
f(81)=1
f(82)=1777
f(83)=1
f(84)=20231
f(85)=643
f(86)=1
f(87)=2659
f(88)=3089
f(89)=1
f(90)=1
f(91)=709
f(92)=1213
f(93)=1
f(94)=2161
f(95)=431
f(96)=229
f(97)=3109
f(98)=25243
f(99)=1601

b) Substitution of the polynom
The polynom f(x)=x^2+176x-1609  could be written as f(y)= y^2-9353 with x=y-88

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+88
f'(x)>2x+175

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 : 1609, 179, 7, 67, 127, 11, 47, 41, 137, 1, 251, 1, 647, 53, 1051, 157, 19, 1, 269, 131,
Found in Database : 1609, 179, 7, 67, 127, 11, 47, 41, 137, 251, 647, 53, 1051, 157, 19, 269, 131, 2311, 79, 3191, 61, 3643, 373, 271, 653, 601, 103, 661, 5531, 317, 593,
Found in Database : 7, 11, 19, 41, 47, 53, 61, 67, 73, 79, 89, 103, 107, 109, 113, 127, 131, 137,