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Development of Algorithmic Constructions

16:24:28 20.Apr 2024

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Polynom = x^2-41x+17

1. Algorithm

a:=1;
b:=-41;
c:=17;


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^2-41x+17

f(0)=17
f(1)=23
f(2)=61
f(3)=97
f(4)=131
f(5)=163
f(6)=193
f(7)=13
f(8)=19
f(9)=271
f(10)=293
f(11)=313
f(12)=331
f(13)=347
f(14)=1
f(15)=373
f(16)=383
f(17)=1
f(18)=397
f(19)=401
f(20)=31
f(21)=1
f(22)=1
f(23)=1
f(24)=1
f(25)=1
f(26)=1
f(27)=1
f(28)=1
f(29)=1
f(30)=1
f(31)=1
f(32)=1
f(33)=1
f(34)=1
f(35)=1
f(36)=1
f(37)=1
f(38)=1
f(39)=1
f(40)=1
f(41)=1
f(42)=59
f(43)=103
f(44)=149
f(45)=197
f(46)=1
f(47)=1
f(48)=353
f(49)=409
f(50)=467
f(51)=1
f(52)=1
f(53)=653
f(54)=719
f(55)=787
f(56)=857
f(57)=929
f(58)=1
f(59)=83
f(60)=89
f(61)=1237
f(62)=1319
f(63)=1
f(64)=1489
f(65)=1
f(66)=1667
f(67)=1759
f(68)=109
f(69)=1949
f(70)=1
f(71)=113
f(72)=173
f(73)=181
f(74)=2459
f(75)=151
f(76)=2677
f(77)=2789
f(78)=2903
f(79)=3019
f(80)=3137
f(81)=3257
f(82)=1
f(83)=1
f(84)=191
f(85)=1
f(86)=1
f(87)=4019
f(88)=4153
f(89)=4289
f(90)=233
f(91)=4567
f(92)=277
f(93)=211
f(94)=4999
f(95)=5147
f(96)=5297
f(97)=5449
f(98)=431
f(99)=443

b) Substitution of the polynom
The polynom f(x)=x^2-41x+17  could be written as f(y)= y^2-403.25 with x=y+20.5

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-20.5
f'(x)>2x-42  with x > 20

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 : 17, 23, 61, 97, 131, 163, 193, 13, 19, 271, 293, 313, 331, 347, 1, 373, 383, 1, 397, 401,
Found in Database : 17, 23, 61, 97, 131, 163, 193, 13, 19, 271, 293, 313, 331, 347, 373, 383, 397, 401, 31,
Found in Database : 13, 17, 19, 23, 31, 59, 61, 83, 89, 97, 103, 109, 113, 131, 149,