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Development of Algorithmic Constructions

23:42:52 19.Apr 2024

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Polynom = x^2+296x-89

1. Algorithm

a:=1;
b:=296;
c:=-89;


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+296x-89

f(0)=89
f(1)=13
f(2)=3
f(3)=101
f(4)=11
f(5)=59
f(6)=1723
f(7)=127
f(8)=71
f(9)=83
f(10)=2971
f(11)=137
f(12)=3607
f(13)=491
f(14)=109
f(15)=1
f(16)=4903
f(17)=1
f(18)=5563
f(19)=67
f(20)=31
f(21)=821
f(22)=6907
f(23)=151
f(24)=7591
f(25)=1
f(26)=251
f(27)=1
f(28)=691
f(29)=389
f(30)=881
f(31)=157
f(32)=3469
f(33)=673
f(34)=11131
f(35)=479
f(36)=11863
f(37)=139
f(38)=4201
f(39)=811
f(40)=79
f(41)=1
f(42)=14107
f(43)=1811
f(44)=4957
f(45)=1907
f(46)=15643
f(47)=167
f(48)=1493
f(49)=1051
f(50)=5737
f(51)=1
f(52)=1637
f(53)=1
f(54)=1447
f(55)=1201
f(56)=211
f(57)=313
f(58)=20443
f(59)=1
f(60)=239
f(61)=2711
f(62)=7369
f(63)=1
f(64)=1
f(65)=487
f(66)=1831
f(67)=233
f(68)=8221
f(69)=3137
f(70)=1
f(71)=541
f(72)=26407
f(73)=839
f(74)=827
f(75)=3467
f(76)=28183
f(77)=1193
f(78)=229
f(79)=1
f(80)=769
f(81)=173
f(82)=997
f(83)=1307
f(84)=1
f(85)=367
f(86)=163
f(87)=1
f(88)=33703
f(89)=1
f(90)=34651
f(91)=4391
f(92)=1
f(93)=347
f(94)=36571
f(95)=193
f(96)=3413
f(97)=2377
f(98)=12841
f(99)=4877

b) Substitution of the polynom
The polynom f(x)=x^2+296x-89  could be written as f(y)= y^2-21993 with x=y-148

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+148
f'(x)>2x+295

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 : 89, 13, 3, 101, 11, 59, 1723, 127, 71, 83, 2971, 137, 3607, 491, 109, 1, 4903, 1, 5563, 67,
Found in Database : 89, 13, 3, 101, 11, 59, 1723, 127, 71, 83, 2971, 137, 3607, 491, 109, 4903, 5563, 67, 31, 821, 6907, 151, 7591, 251, 691, 389, 881, 157, 3469, 673, 11131, 479, 11863, 139, 4201, 811,
Found in Database : 3, 11, 13, 31, 59, 67, 71, 79, 83, 89, 101, 109, 127, 137, 139,