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Development of Algorithmic Constructions

00:32:18 20.Apr 2024

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Polynom = x^2+204x-397

1. Algorithm

a:=1;
b:=204;
c:=-397;


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-397

f(0)=397
f(1)=3
f(2)=5
f(3)=7
f(4)=29
f(5)=1
f(6)=863
f(7)=1
f(8)=433
f(9)=19
f(10)=83
f(11)=41
f(12)=439
f(13)=101
f(14)=59
f(15)=1
f(16)=347
f(17)=1
f(18)=61
f(19)=1
f(20)=1361
f(21)=541
f(22)=1
f(23)=67
f(24)=1
f(25)=37
f(26)=1861
f(27)=73
f(28)=107
f(29)=53
f(30)=179
f(31)=1
f(32)=1
f(33)=1
f(34)=1
f(35)=1
f(36)=8243
f(37)=71
f(38)=419
f(39)=227
f(40)=3121
f(41)=1
f(42)=1987
f(43)=1
f(44)=701
f(45)=193
f(46)=3701
f(47)=1
f(48)=11699
f(49)=1
f(50)=1367
f(51)=197
f(52)=1
f(53)=1
f(54)=2707
f(55)=577
f(56)=4721
f(57)=181
f(58)=4933
f(59)=1
f(60)=15443
f(61)=1
f(62)=1
f(63)=2053
f(64)=1117
f(65)=89
f(66)=131
f(67)=1
f(68)=2011
f(69)=461
f(70)=2087
f(71)=797
f(72)=1
f(73)=1
f(74)=269
f(75)=1283
f(76)=6961
f(77)=1
f(78)=21599
f(79)=1
f(80)=1063
f(81)=709
f(82)=1
f(83)=1
f(84)=4759
f(85)=1
f(86)=1
f(87)=1
f(88)=937
f(89)=1
f(90)=389
f(91)=1
f(92)=1789
f(93)=1
f(94)=263
f(95)=1
f(96)=28403
f(97)=1
f(98)=9733
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+204x-397  could be written as f(y)= y^2-10801 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+203

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 : 397, 3, 5, 7, 29, 1, 863, 1, 433, 19, 83, 41, 439, 101, 59, 1, 347, 1, 61, 1,
Found in Database : 397, 3, 5, 7, 29, 863, 433, 19, 83, 41, 439, 101, 59, 347, 61, 1361, 541, 67, 37, 1861, 73, 107, 53, 179, 8243, 71, 419, 227,
Found in Database : 3, 5, 7, 19, 29, 37, 41, 53, 59, 61, 67, 71, 73, 83, 89, 101, 107, 131, 137, 149,