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Development of Algorithmic Constructions

21:21:37 16.Apr 2024

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Polynom = x^2+23x-47

1. Algorithm

a:=1;
b:=23;
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+23x-47

f(0)=47
f(1)=23
f(2)=3
f(3)=31
f(4)=61
f(5)=1
f(6)=127
f(7)=163
f(8)=67
f(9)=241
f(10)=283
f(11)=109
f(12)=373
f(13)=421
f(14)=157
f(15)=523
f(16)=577
f(17)=211
f(18)=691
f(19)=751
f(20)=271
f(21)=877
f(22)=41
f(23)=337
f(24)=1
f(25)=1153
f(26)=409
f(27)=1303
f(28)=1381
f(29)=487
f(30)=1543
f(31)=1627
f(32)=571
f(33)=1801
f(34)=1
f(35)=661
f(36)=1
f(37)=53
f(38)=757
f(39)=2371
f(40)=2473
f(41)=859
f(42)=2683
f(43)=2791
f(44)=967
f(45)=131
f(46)=59
f(47)=1
f(48)=3361
f(49)=1
f(50)=1201
f(51)=3727
f(52)=3853
f(53)=1327
f(54)=4111
f(55)=4243
f(56)=1459
f(57)=4513
f(58)=4651
f(59)=1597
f(60)=4933
f(61)=5077
f(62)=1741
f(63)=1
f(64)=5521
f(65)=1
f(66)=5827
f(67)=193
f(68)=89
f(69)=6301
f(70)=281
f(71)=1
f(72)=6793
f(73)=6961
f(74)=2377
f(75)=1
f(76)=7477
f(77)=2551
f(78)=191
f(79)=8011
f(80)=2731
f(81)=8377
f(82)=8563
f(83)=2917
f(84)=8941
f(85)=9133
f(86)=3109
f(87)=107
f(88)=9721
f(89)=3307
f(90)=1
f(91)=449
f(92)=3511
f(93)=467
f(94)=233
f(95)=1
f(96)=367
f(97)=11593
f(98)=1
f(99)=227

b) Substitution of the polynom
The polynom f(x)=x^2+23x-47  could be written as f(y)= y^2-179.25 with x=y-11.5

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+11.5
f'(x)>2x+22

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, 23, 3, 31, 61, 1, 127, 163, 67, 241, 283, 109, 373, 421, 157, 523, 577, 211, 691, 751,
Found in Database : 47, 23, 3, 31, 61, 127, 163, 67, 241, 283, 109, 373, 421, 157, 523, 577, 211, 691, 751, 271, 877, 41, 337, 1153, 409, 1303, 1381, 487, 1543, 1627, 571, 1801, 661, 53, 757, 2371,
Found in Database : 3, 23, 31, 41, 47, 53, 59, 61, 67, 89, 107, 109, 127, 131, 137, 149,