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Development of Algorithmic Constructions

18:46:27 19.Apr 2024

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Polynom = x^2+44x-421

1. Algorithm

a:=1;
b:=44;
c:=-421;


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+44x-421

f(0)=421
f(1)=47
f(2)=7
f(3)=5
f(4)=229
f(5)=11
f(6)=1
f(7)=1
f(8)=1
f(9)=1
f(10)=17
f(11)=23
f(12)=251
f(13)=1
f(14)=1
f(15)=29
f(16)=1
f(17)=1
f(18)=139
f(19)=97
f(20)=859
f(21)=59
f(22)=1031
f(23)=1
f(24)=173
f(25)=163
f(26)=1399
f(27)=1
f(28)=1
f(29)=53
f(30)=257
f(31)=1
f(32)=2011
f(33)=1
f(34)=1
f(35)=293
f(36)=2459
f(37)=1
f(38)=1
f(39)=1
f(40)=2939
f(41)=383
f(42)=3191
f(43)=83
f(44)=1
f(45)=1
f(46)=3719
f(47)=241
f(48)=1
f(49)=1
f(50)=389
f(51)=79
f(52)=653
f(53)=1
f(54)=4871
f(55)=157
f(56)=5179
f(57)=1
f(58)=1
f(59)=101
f(60)=1
f(61)=1
f(62)=6151
f(63)=1
f(64)=6491
f(65)=1
f(66)=977
f(67)=877
f(68)=1439
f(69)=461
f(70)=7559
f(71)=1
f(72)=103
f(73)=1
f(74)=8311
f(75)=1063
f(76)=8699
f(77)=1
f(78)=107
f(79)=1
f(80)=1
f(81)=1213
f(82)=1
f(83)=1
f(84)=10331
f(85)=659
f(86)=1
f(87)=1
f(88)=2239
f(89)=1427
f(90)=113
f(91)=1483
f(92)=1
f(93)=1
f(94)=1
f(95)=1
f(96)=277
f(97)=1657
f(98)=2699
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+44x-421  could be written as f(y)= y^2-905 with x=y-22

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+22
f'(x)>2x+43

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 : 421, 47, 7, 5, 229, 11, 1, 1, 1, 1, 17, 23, 251, 1, 1, 29, 1, 1, 139, 97,
Found in Database : 421, 47, 7, 5, 229, 11, 17, 23, 251, 29, 139, 97, 859, 59, 1031, 173, 163, 1399, 53, 257, 2011, 293, 2459,
Found in Database : 5, 7, 11, 17, 23, 29, 47, 53, 59, 79, 83, 97, 101, 103, 107, 113, 127, 139,