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Development of Algorithmic Constructions

20:43:34 19.Apr 2024

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Polynom = x^2+6x-113

1. Algorithm

a:=1;
b:=6;
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+6x-113

f(0)=113
f(1)=53
f(2)=97
f(3)=43
f(4)=73
f(5)=29
f(6)=41
f(7)=11
f(8)=1
f(9)=1
f(10)=47
f(11)=37
f(12)=103
f(13)=67
f(14)=167
f(15)=101
f(16)=239
f(17)=139
f(18)=1
f(19)=181
f(20)=1
f(21)=227
f(22)=503
f(23)=277
f(24)=607
f(25)=331
f(26)=719
f(27)=389
f(28)=839
f(29)=1
f(30)=967
f(31)=1
f(32)=1103
f(33)=587
f(34)=1
f(35)=661
f(36)=1399
f(37)=739
f(38)=1559
f(39)=821
f(40)=157
f(41)=907
f(42)=173
f(43)=997
f(44)=2087
f(45)=1091
f(46)=1
f(47)=1
f(48)=1
f(49)=1291
f(50)=2687
f(51)=127
f(52)=2903
f(53)=137
f(54)=59
f(55)=1621
f(56)=3359
f(57)=1
f(58)=61
f(59)=1861
f(60)=3847
f(61)=1987
f(62)=373
f(63)=1
f(64)=397
f(65)=2251
f(66)=4639
f(67)=2389
f(68)=4919
f(69)=2531
f(70)=1
f(71)=2677
f(72)=5503
f(73)=257
f(74)=5807
f(75)=271
f(76)=211
f(77)=1
f(78)=1
f(79)=3301
f(80)=1
f(81)=3467
f(82)=7103
f(83)=3637
f(84)=677
f(85)=1
f(86)=709
f(87)=3989
f(88)=199
f(89)=1
f(90)=8527
f(91)=4357
f(92)=307
f(93)=4547
f(94)=251
f(95)=431
f(96)=9679
f(97)=449
f(98)=10079
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+6x-113  could be written as f(y)= y^2-122 with x=y-3

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+3
f'(x)>2x+5

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, 53, 97, 43, 73, 29, 41, 11, 1, 1, 47, 37, 103, 67, 167, 101, 239, 139, 1, 181,
Found in Database : 113, 53, 97, 43, 73, 29, 41, 11, 47, 37, 103, 67, 167, 101, 239, 139, 181, 227, 503, 277, 607, 331, 719, 389, 839, 967, 1103, 587, 661, 1399, 739, 1559, 821,
Found in Database : 11, 29, 37, 41, 43, 47, 53, 59, 61, 67, 73, 97, 101, 103, 113, 127, 137, 139,