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Development of Algorithmic Constructions

14:18:04 19.Apr 2024

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Polynom = x^2+246x-631

1. Algorithm

a:=1;
b:=246;
c:=-631;


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+246x-631

f(0)=631
f(1)=3
f(2)=5
f(3)=29
f(4)=41
f(5)=13
f(6)=881
f(7)=19
f(8)=467
f(9)=1
f(10)=643
f(11)=61
f(12)=17
f(13)=1
f(14)=59
f(15)=821
f(16)=1187
f(17)=1
f(18)=317
f(19)=367
f(20)=521
f(21)=311
f(22)=1
f(23)=463
f(24)=5849
f(25)=1
f(26)=113
f(27)=337
f(28)=2347
f(29)=1
f(30)=7649
f(31)=1
f(32)=1
f(33)=67
f(34)=2963
f(35)=1
f(36)=9521
f(37)=1
f(38)=1129
f(39)=2621
f(40)=1201
f(41)=1
f(42)=2293
f(43)=983
f(44)=1
f(45)=1
f(46)=251
f(47)=73
f(48)=1
f(49)=1
f(50)=4723
f(51)=191
f(52)=991
f(53)=1
f(54)=15569
f(55)=1327
f(56)=1
f(57)=1
f(58)=1889
f(59)=1447
f(60)=17729
f(61)=1
f(62)=1231
f(63)=277
f(64)=1
f(65)=1
f(66)=19961
f(67)=1
f(68)=6907
f(69)=1319
f(70)=1
f(71)=1823
f(72)=1
f(73)=1
f(74)=197
f(75)=5861
f(76)=883
f(77)=101
f(78)=601
f(79)=2087
f(80)=499
f(81)=1
f(82)=103
f(83)=1
f(84)=263
f(85)=1
f(86)=227
f(87)=109
f(88)=9587
f(89)=1
f(90)=1021
f(91)=2503
f(92)=677
f(93)=1931
f(94)=1
f(95)=2647
f(96)=2477
f(97)=1
f(98)=11027
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+246x-631  could be written as f(y)= y^2-15760 with x=y-123

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+123
f'(x)>2x+245

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 : 631, 3, 5, 29, 41, 13, 881, 19, 467, 1, 643, 61, 17, 1, 59, 821, 1187, 1, 317, 367,
Found in Database : 631, 3, 5, 29, 41, 13, 881, 19, 467, 643, 61, 17, 59, 821, 1187, 317, 367, 521, 311, 463, 5849, 113, 337, 2347, 7649, 67, 2963, 9521, 1129, 2621,
Found in Database : 3, 5, 13, 17, 19, 29, 41, 59, 61, 67, 73, 101, 103, 109, 113,