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Development of Algorithmic Constructions

23:19:42 18.Apr 2024

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Polynom = x^2+1x-73

1. Algorithm

a:=1;
b:=1;
c:=-73;


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+1x-73

f(0)=73
f(1)=71
f(2)=67
f(3)=61
f(4)=53
f(5)=43
f(6)=31
f(7)=17
f(8)=1
f(9)=1
f(10)=37
f(11)=59
f(12)=83
f(13)=109
f(14)=137
f(15)=167
f(16)=199
f(17)=233
f(18)=269
f(19)=307
f(20)=347
f(21)=389
f(22)=433
f(23)=479
f(24)=1
f(25)=577
f(26)=1
f(27)=683
f(28)=739
f(29)=797
f(30)=857
f(31)=919
f(32)=983
f(33)=1049
f(34)=1117
f(35)=1187
f(36)=1259
f(37)=1
f(38)=1409
f(39)=1487
f(40)=1567
f(41)=97
f(42)=1733
f(43)=107
f(44)=1907
f(45)=1997
f(46)=2089
f(47)=1
f(48)=1
f(49)=2377
f(50)=2477
f(51)=2579
f(52)=2683
f(53)=2789
f(54)=2897
f(55)=1
f(56)=3119
f(57)=1
f(58)=197
f(59)=3467
f(60)=211
f(61)=3709
f(62)=3833
f(63)=1
f(64)=1
f(65)=4217
f(66)=4349
f(67)=4483
f(68)=149
f(69)=1
f(70)=1
f(71)=5039
f(72)=1
f(73)=1
f(74)=5477
f(75)=331
f(76)=5779
f(77)=349
f(78)=6089
f(79)=6247
f(80)=1
f(81)=6569
f(82)=6733
f(83)=6899
f(84)=191
f(85)=7237
f(86)=239
f(87)=7583
f(88)=7759
f(89)=7937
f(90)=8117
f(91)=193
f(92)=499
f(93)=8669
f(94)=521
f(95)=1
f(96)=9239
f(97)=9433
f(98)=9629
f(99)=317

b) Substitution of the polynom
The polynom f(x)=x^2+1x-73  could be written as f(y)= y^2-73.25 with x=y-0.5

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+0.5
f'(x)>2x

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 : 73, 71, 67, 61, 53, 43, 31, 17, 1, 1, 37, 59, 83, 109, 137, 167, 199, 233, 269, 307,
Found in Database : 73, 71, 67, 61, 53, 43, 31, 17, 37, 59, 83, 109, 137, 167, 199, 233, 269, 307, 347, 389, 433, 479, 577, 683, 739, 797, 857, 919, 983, 1049, 1117, 1187, 1259, 1409, 1487,
Found in Database : 17, 31, 37, 43, 53, 59, 61, 67, 71, 73, 83, 97, 107, 109, 137, 149,