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Development of Algorithmic Constructions

21:02:02 18.Apr 2024

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Polynom = x^2+92x-113

1. Algorithm

a:=1;
b:=92;
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+92x-113

f(0)=113
f(1)=5
f(2)=3
f(3)=43
f(4)=271
f(5)=31
f(6)=19
f(7)=29
f(8)=229
f(9)=199
f(10)=907
f(11)=17
f(12)=227
f(13)=313
f(14)=457
f(15)=373
f(16)=1
f(17)=1
f(18)=1867
f(19)=499
f(20)=709
f(21)=1
f(22)=479
f(23)=211
f(24)=2671
f(25)=37
f(26)=197
f(27)=1
f(28)=191
f(29)=283
f(30)=3547
f(31)=1
f(32)=257
f(33)=59
f(34)=97
f(35)=1
f(36)=1
f(37)=233
f(38)=1609
f(39)=1249
f(40)=5167
f(41)=89
f(42)=1103
f(43)=1423
f(44)=103
f(45)=1
f(46)=1
f(47)=107
f(48)=6607
f(49)=1699
f(50)=137
f(51)=359
f(52)=1
f(53)=631
f(54)=409
f(55)=1993
f(56)=109
f(57)=419
f(58)=277
f(59)=733
f(60)=9007
f(61)=461
f(62)=1
f(63)=127
f(64)=9871
f(65)=1
f(66)=2063
f(67)=1
f(68)=1
f(69)=2749
f(70)=1
f(71)=1
f(72)=2339
f(73)=157
f(74)=4057
f(75)=1
f(76)=2531
f(77)=1
f(78)=13147
f(79)=1
f(80)=4549
f(81)=139
f(82)=149
f(83)=1201
f(84)=863
f(85)=3733
f(86)=1013
f(87)=773
f(88)=15727
f(89)=1
f(90)=16267
f(91)=827
f(92)=1
f(93)=4273
f(94)=599
f(95)=1471
f(96)=1
f(97)=911
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+92x-113  could be written as f(y)= y^2-2229 with x=y-46

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+46
f'(x)>2x+91

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, 3, 43, 271, 31, 19, 29, 229, 199, 907, 17, 227, 313, 457, 373, 1, 1, 1867, 499,
Found in Database : 113, 5, 3, 43, 271, 31, 19, 29, 229, 199, 907, 17, 227, 313, 457, 373, 1867, 499, 709, 479, 211, 2671, 37, 197, 191, 283, 3547, 257, 59, 97, 233, 1609, 1249,
Found in Database : 3, 5, 17, 19, 29, 31, 37, 43, 59, 89, 97, 103, 107, 109, 113, 127, 137, 139, 149,