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Development of Algorithmic Constructions

09:31:25 19.Apr 2024

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Polynom = x^2+3x-7

1. Algorithm

a:=1;
b:=3;
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+3x-7

f(0)=7
f(1)=3
f(2)=1
f(3)=11
f(4)=1
f(5)=1
f(6)=47
f(7)=1
f(8)=1
f(9)=101
f(10)=41
f(11)=1
f(12)=173
f(13)=67
f(14)=1
f(15)=263
f(16)=1
f(17)=37
f(18)=53
f(19)=137
f(20)=151
f(21)=71
f(22)=181
f(23)=197
f(24)=641
f(25)=1
f(26)=83
f(27)=73
f(28)=1
f(29)=307
f(30)=983
f(31)=349
f(32)=1
f(33)=1181
f(34)=139
f(35)=1
f(36)=127
f(37)=491
f(38)=1
f(39)=233
f(40)=571
f(41)=599
f(42)=269
f(43)=1
f(44)=229
f(45)=2153
f(46)=107
f(47)=1
f(48)=2441
f(49)=1
f(50)=881
f(51)=1
f(52)=317
f(53)=1
f(54)=1
f(55)=1061
f(56)=157
f(57)=3413
f(58)=1
f(59)=1217
f(60)=1
f(61)=433
f(62)=149
f(63)=593
f(64)=1427
f(65)=1471
f(66)=4547
f(67)=223
f(68)=1607
f(69)=1
f(70)=1
f(71)=1
f(72)=5393
f(73)=1847
f(74)=271
f(75)=5843
f(76)=1999
f(77)=293
f(78)=6311
f(79)=719
f(80)=1
f(81)=971
f(82)=211
f(83)=2377
f(84)=1
f(85)=1
f(86)=2549
f(87)=7823
f(88)=1
f(89)=1
f(90)=8363
f(91)=1
f(92)=1
f(93)=811
f(94)=3037
f(95)=443
f(96)=9497
f(97)=359
f(98)=1
f(99)=10091

b) Substitution of the polynom
The polynom f(x)=x^2+3x-7  could be written as f(y)= y^2-9.25 with x=y-1.5

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+1.5
f'(x)>2x+2

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, 3, 1, 11, 1, 1, 47, 1, 1, 101, 41, 1, 173, 67, 1, 263, 1, 37, 53, 137,
Found in Database : 7, 3, 11, 47, 101, 41, 173, 67, 263, 37, 53, 137, 151, 71, 181, 197, 641, 83, 73, 307, 983, 349, 1181, 139, 127, 491, 233,
Found in Database : 3, 7, 11, 37, 41, 47, 53, 67, 71, 73, 83, 101, 107, 127, 137, 139, 149,