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Development of Algorithmic Constructions

23:22:39 19.Apr 2024

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Polynom = x^2+3x-47

1. Algorithm

a:=1;
b:=3;
c:=-47;


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+3x-47

f(0)=47
f(1)=43
f(2)=37
f(3)=29
f(4)=19
f(5)=7
f(6)=1
f(7)=23
f(8)=41
f(9)=61
f(10)=83
f(11)=107
f(12)=1
f(13)=1
f(14)=191
f(15)=223
f(16)=257
f(17)=293
f(18)=331
f(19)=53
f(20)=59
f(21)=457
f(22)=503
f(23)=1
f(24)=601
f(25)=653
f(26)=101
f(27)=109
f(28)=821
f(29)=881
f(30)=1
f(31)=1
f(32)=1
f(33)=163
f(34)=173
f(35)=1283
f(36)=1
f(37)=1433
f(38)=1511
f(39)=1
f(40)=239
f(41)=251
f(42)=97
f(43)=1931
f(44)=1
f(45)=2113
f(46)=2207
f(47)=1
f(48)=1
f(49)=1
f(50)=137
f(51)=2707
f(52)=1
f(53)=127
f(54)=433
f(55)=449
f(56)=3257
f(57)=3373
f(58)=3491
f(59)=157
f(60)=3733
f(61)=1
f(62)=569
f(63)=4111
f(64)=4241
f(65)=4373
f(66)=4507
f(67)=4643
f(68)=683
f(69)=1
f(70)=1
f(71)=1
f(72)=1
f(73)=5501
f(74)=5651
f(75)=829
f(76)=1
f(77)=6113
f(78)=6271
f(79)=1
f(80)=347
f(81)=233
f(82)=1
f(83)=1013
f(84)=1
f(85)=7433
f(86)=7607
f(87)=181
f(88)=419
f(89)=1163
f(90)=1
f(91)=1
f(92)=8693
f(93)=1
f(94)=193
f(95)=1
f(96)=1
f(97)=197
f(98)=9851
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+3x-47  could be written as f(y)= y^2-49.25 with x=y-1.5

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+1.5
f'(x)>2x+2

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 : 47, 43, 37, 29, 19, 7, 1, 23, 41, 61, 83, 107, 1, 1, 191, 223, 257, 293, 331, 53,
Found in Database : 47, 43, 37, 29, 19, 7, 23, 41, 61, 83, 107, 191, 223, 257, 293, 331, 53, 59, 457, 503, 601, 653, 101, 109, 821, 881, 163, 173, 1283, 1433, 1511,
Found in Database : 7, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, 83, 97, 101, 107, 109, 127, 137,