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Development of Algorithmic Constructions

03:10:12 18.Apr 2024

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Polynom = x^2-40x-7

1. Algorithm

a:=1;
b:=-40;
c:=-7;


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^2-40x-7

f(0)=7
f(1)=23
f(2)=83
f(3)=59
f(4)=151
f(5)=13
f(6)=211
f(7)=17
f(8)=263
f(9)=11
f(10)=307
f(11)=163
f(12)=1
f(13)=179
f(14)=53
f(15)=191
f(16)=1
f(17)=199
f(18)=31
f(19)=29
f(20)=37
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)=1
f(43)=61
f(44)=1
f(45)=109
f(46)=269
f(47)=1
f(48)=1
f(49)=1
f(50)=1
f(51)=277
f(52)=617
f(53)=1
f(54)=107
f(55)=409
f(56)=127
f(57)=1
f(58)=1
f(59)=557
f(60)=1193
f(61)=1
f(62)=1
f(63)=103
f(64)=139
f(65)=809
f(66)=1709
f(67)=1
f(68)=271
f(69)=997
f(70)=1
f(71)=1097
f(72)=2297
f(73)=1201
f(74)=193
f(75)=1
f(76)=2729
f(77)=1
f(78)=2957
f(79)=1
f(80)=1
f(81)=1657
f(82)=491
f(83)=137
f(84)=1
f(85)=1
f(86)=359
f(87)=157
f(88)=4217
f(89)=311
f(90)=4493
f(91)=331
f(92)=281
f(93)=1
f(94)=1
f(95)=2609
f(96)=1
f(97)=251
f(98)=811
f(99)=2917

b) Substitution of the polynom
The polynom f(x)=x^2-40x-7  could be written as f(y)= y^2-407 with x=y+20

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
f'(x)>2x-41  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 : 7, 23, 83, 59, 151, 13, 211, 17, 263, 11, 307, 163, 1, 179, 53, 191, 1, 199, 31, 29,
Found in Database : 7, 23, 83, 59, 151, 13, 211, 17, 263, 11, 307, 163, 179, 53, 191, 199, 31, 29, 37,
Found in Database : 7, 11, 13, 17, 23, 29, 31, 37, 53, 59, 61, 83, 103, 107, 109, 127, 137, 139,