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Development of Algorithmic Constructions

11:46:54 20.Apr 2024

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Polynom = x^2-204x+971

1. Algorithm

a:=1;
b:=-204;
c:=971;


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^2-204x+971

f(0)=971
f(1)=3
f(2)=7
f(3)=23
f(4)=19
f(5)=1
f(6)=31
f(7)=17
f(8)=199
f(9)=1
f(10)=1
f(11)=1
f(12)=43
f(13)=1
f(14)=563
f(15)=233
f(16)=97
f(17)=1
f(18)=2377
f(19)=53
f(20)=1
f(21)=359
f(22)=337
f(23)=1
f(24)=197
f(25)=73
f(26)=1
f(27)=1
f(28)=1319
f(29)=1
f(30)=607
f(31)=61
f(32)=1511
f(33)=1
f(34)=229
f(35)=103
f(36)=5077
f(37)=1
f(38)=593
f(39)=683
f(40)=1
f(41)=1
f(42)=307
f(43)=1
f(44)=1
f(45)=773
f(46)=2099
f(47)=89
f(48)=1
f(49)=1
f(50)=2243
f(51)=1
f(52)=2311
f(53)=293
f(54)=7129
f(55)=1
f(56)=271
f(57)=463
f(58)=1
f(59)=79
f(60)=7669
f(61)=1
f(62)=373
f(63)=1
f(64)=2663
f(65)=1
f(66)=1
f(67)=1
f(68)=1
f(69)=149
f(70)=2803
f(71)=353
f(72)=1
f(73)=179
f(74)=1
f(75)=1
f(76)=139
f(77)=367
f(78)=521
f(79)=1
f(80)=157
f(81)=281
f(82)=3011
f(83)=1
f(84)=9109
f(85)=127
f(86)=1
f(87)=1151
f(88)=3079
f(89)=193
f(90)=1327
f(91)=1
f(92)=1
f(93)=167
f(94)=347
f(95)=1
f(96)=9397
f(97)=1
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-204x+971  could be written as f(y)= y^2-9433 with x=y+102

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-102
f'(x)>2x-205  with x > 97

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 : 971, 3, 7, 23, 19, 1, 31, 17, 199, 1, 1, 1, 43, 1, 563, 233, 97, 1, 2377, 53,
Found in Database : 971, 3, 7, 23, 19, 31, 17, 199, 43, 563, 233, 97, 2377, 53, 359, 337, 197, 73, 1319, 607, 61, 1511, 229, 103, 5077, 593, 683,
Found in Database : 3, 7, 17, 19, 23, 31, 43, 53, 61, 73, 79, 89, 97, 103, 127, 131, 139, 149,