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Development of Algorithmic Constructions

22:22:42 18.Apr 2024

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Polynom = x^2+32x-127

1. Algorithm

a:=1;
b:=32;
c:=-127;


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+32x-127

f(0)=127
f(1)=47
f(2)=59
f(3)=11
f(4)=17
f(5)=29
f(6)=101
f(7)=73
f(8)=193
f(9)=1
f(10)=293
f(11)=173
f(12)=401
f(13)=229
f(14)=1
f(15)=1
f(16)=641
f(17)=353
f(18)=773
f(19)=421
f(20)=83
f(21)=1
f(22)=1061
f(23)=569
f(24)=1217
f(25)=1
f(26)=1381
f(27)=733
f(28)=1553
f(29)=821
f(30)=1733
f(31)=1
f(32)=113
f(33)=1009
f(34)=1
f(35)=1109
f(36)=211
f(37)=1213
f(38)=149
f(39)=1321
f(40)=2753
f(41)=1433
f(42)=271
f(43)=1549
f(44)=3217
f(45)=1669
f(46)=3461
f(47)=163
f(48)=79
f(49)=1
f(50)=137
f(51)=2053
f(52)=4241
f(53)=199
f(54)=4517
f(55)=1
f(56)=4801
f(57)=2473
f(58)=463
f(59)=2621
f(60)=5393
f(61)=1
f(62)=5701
f(63)=1
f(64)=547
f(65)=3089
f(66)=373
f(67)=3253
f(68)=6673
f(69)=311
f(70)=7013
f(71)=3593
f(72)=433
f(73)=3769
f(74)=7717
f(75)=359
f(76)=8081
f(77)=4133
f(78)=107
f(79)=1
f(80)=1
f(81)=4513
f(82)=9221
f(83)=277
f(84)=1
f(85)=4909
f(86)=911
f(87)=5113
f(88)=10433
f(89)=313
f(90)=10853
f(91)=503
f(92)=389
f(93)=5749
f(94)=11717
f(95)=1
f(96)=12161
f(97)=563
f(98)=12613
f(99)=6421

b) Substitution of the polynom
The polynom f(x)=x^2+32x-127  could be written as f(y)= y^2-383 with x=y-16

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+16
f'(x)>2x+31

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 : 127, 47, 59, 11, 17, 29, 101, 73, 193, 1, 293, 173, 401, 229, 1, 1, 641, 353, 773, 421,
Found in Database : 127, 47, 59, 11, 17, 29, 101, 73, 193, 293, 173, 401, 229, 641, 353, 773, 421, 83, 1061, 569, 1217, 1381, 733, 1553, 821, 1733, 113, 1009, 1109, 211, 1213, 149, 1321,
Found in Database : 11, 17, 29, 47, 59, 73, 79, 83, 101, 107, 113, 127, 131, 137, 149,