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Development of Algorithmic Constructions

17:36:44 20.Apr 2024

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Polynom = x^2+28x-53

1. Algorithm

a:=1;
b:=28;
c:=-53;


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+28x-53

f(0)=53
f(1)=3
f(2)=7
f(3)=5
f(4)=1
f(5)=1
f(6)=151
f(7)=1
f(8)=47
f(9)=1
f(10)=109
f(11)=1
f(12)=61
f(13)=1
f(14)=107
f(15)=37
f(16)=31
f(17)=89
f(18)=1
f(19)=1
f(20)=907
f(21)=1
f(22)=349
f(23)=1
f(24)=239
f(25)=1
f(26)=193
f(27)=179
f(28)=101
f(29)=1
f(30)=241
f(31)=1
f(32)=1867
f(33)=1
f(34)=137
f(35)=269
f(36)=2251
f(37)=1
f(38)=491
f(39)=1
f(40)=127
f(41)=347
f(42)=2887
f(43)=1
f(44)=1
f(45)=1
f(46)=1117
f(47)=1
f(48)=719
f(49)=1
f(50)=3847
f(51)=71
f(52)=1
f(53)=1
f(54)=1
f(55)=1
f(56)=4651
f(57)=599
f(58)=1
f(59)=1
f(60)=5227
f(61)=1
f(62)=5527
f(63)=1
f(64)=389
f(65)=1
f(66)=6151
f(67)=263
f(68)=1
f(69)=83
f(70)=2269
f(71)=1
f(72)=1021
f(73)=1
f(74)=1499
f(75)=1
f(76)=2617
f(77)=251
f(78)=1
f(79)=1
f(80)=277
f(81)=1097
f(82)=1
f(83)=229
f(84)=1871
f(85)=199
f(86)=1
f(87)=311
f(88)=677
f(89)=1
f(90)=10567
f(91)=449
f(92)=10987
f(93)=1
f(94)=761
f(95)=727
f(96)=1693
f(97)=503
f(98)=2459
f(99)=313

b) Substitution of the polynom
The polynom f(x)=x^2+28x-53  could be written as f(y)= y^2-249 with x=y-14

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+14
f'(x)>2x+27

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 : 53, 3, 7, 5, 1, 1, 151, 1, 47, 1, 109, 1, 61, 1, 107, 37, 31, 89, 1, 1,
Found in Database : 53, 3, 7, 5, 151, 47, 109, 61, 107, 37, 31, 89, 907, 349, 239, 193, 179, 101, 241, 1867, 137, 269, 2251, 491,
Found in Database : 3, 5, 7, 31, 37, 47, 53, 61, 71, 83, 89, 101, 107, 109, 127, 137, 149,