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Development of Algorithmic Constructions

05:35:32 19.Apr 2024

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Polynom = x^2+72x-113

1. Algorithm

a:=1;
b:=72;
c:=-113;


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+72x-113

f(0)=113
f(1)=5
f(2)=7
f(3)=1
f(4)=191
f(5)=17
f(6)=71
f(7)=11
f(8)=31
f(9)=1
f(10)=101
f(11)=1
f(12)=179
f(13)=1
f(14)=1091
f(15)=149
f(16)=37
f(17)=1
f(18)=137
f(19)=1
f(20)=157
f(21)=23
f(22)=1
f(23)=1
f(24)=313
f(25)=1
f(26)=487
f(27)=1
f(28)=2687
f(29)=1
f(30)=421
f(31)=1
f(32)=643
f(33)=419
f(34)=3491
f(35)=227
f(36)=151
f(37)=1
f(38)=83
f(39)=1
f(40)=397
f(41)=1
f(42)=1
f(43)=1
f(44)=1
f(45)=1
f(46)=1063
f(47)=1
f(48)=5647
f(49)=727
f(50)=5987
f(51)=1
f(52)=181
f(53)=1
f(54)=6691
f(55)=859
f(56)=1
f(57)=1
f(58)=1061
f(59)=1
f(60)=211
f(61)=1
f(62)=1
f(63)=1049
f(64)=1
f(65)=1
f(66)=257
f(67)=1
f(68)=409
f(69)=601
f(70)=317
f(71)=251
f(72)=293
f(73)=1
f(74)=10691
f(75)=1
f(76)=131
f(77)=1
f(78)=11587
f(79)=1
f(80)=1721
f(81)=307
f(82)=2503
f(83)=797
f(84)=1181
f(85)=827
f(86)=1
f(87)=1
f(88)=13967
f(89)=1777
f(90)=1
f(91)=1
f(92)=599
f(93)=1
f(94)=2213
f(95)=1
f(96)=3203
f(97)=1
f(98)=16547
f(99)=1051

b) Substitution of the polynom
The polynom f(x)=x^2+72x-113  could be written as f(y)= y^2-1409 with x=y-36

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+36
f'(x)>2x+71

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 : 113, 5, 7, 1, 191, 17, 71, 11, 31, 1, 101, 1, 179, 1, 1091, 149, 37, 1, 137, 1,
Found in Database : 113, 5, 7, 191, 17, 71, 11, 31, 101, 179, 1091, 149, 37, 137, 157, 23, 313, 487, 2687, 421, 643, 419, 3491, 227, 151, 83,
Found in Database : 5, 7, 11, 17, 23, 31, 37, 71, 83, 101, 113, 131, 137, 149,