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Development of Algorithmic Constructions

11:44:26 20.Apr 2024

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Polynom = x^2+88x-577

1. Algorithm

a:=1;
b:=88;
c:=-577;


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+88x-577

f(0)=577
f(1)=61
f(2)=397
f(3)=19
f(4)=11
f(5)=7
f(6)=13
f(7)=1
f(8)=191
f(9)=37
f(10)=31
f(11)=1
f(12)=89
f(13)=23
f(14)=1
f(15)=1
f(16)=1087
f(17)=151
f(18)=1
f(19)=1
f(20)=1583
f(21)=107
f(22)=97
f(23)=1
f(24)=2111
f(25)=281
f(26)=1
f(27)=79
f(28)=2671
f(29)=1
f(30)=2963
f(31)=389
f(32)=251
f(33)=1
f(34)=3571
f(35)=233
f(36)=1
f(37)=1
f(38)=4211
f(39)=547
f(40)=59
f(41)=1
f(42)=257
f(43)=1
f(44)=5231
f(45)=1
f(46)=1
f(47)=103
f(48)=541
f(49)=1
f(50)=6323
f(51)=1
f(52)=6703
f(53)=431
f(54)=1013
f(55)=911
f(56)=7487
f(57)=1
f(58)=607
f(59)=1
f(60)=1
f(61)=1
f(62)=1
f(63)=1117
f(64)=9151
f(65)=1171
f(66)=9587
f(67)=613
f(68)=1433
f(69)=641
f(70)=953
f(71)=1
f(72)=353
f(73)=127
f(74)=11411
f(75)=1
f(76)=11887
f(77)=379
f(78)=139
f(79)=83
f(80)=677
f(81)=149
f(82)=1
f(83)=1
f(84)=1
f(85)=883
f(86)=14387
f(87)=1831
f(88)=1
f(89)=271
f(90)=15443
f(91)=491
f(92)=1453
f(93)=1
f(94)=1
f(95)=1
f(96)=2441
f(97)=167
f(98)=929
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+88x-577  could be written as f(y)= y^2-2513 with x=y-44

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+44
f'(x)>2x+87

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 : 577, 61, 397, 19, 11, 7, 13, 1, 191, 37, 31, 1, 89, 23, 1, 1, 1087, 151, 1, 1,
Found in Database : 577, 61, 397, 19, 11, 7, 13, 191, 37, 31, 89, 23, 1087, 151, 1583, 107, 97, 2111, 281, 79, 2671, 2963, 389, 251, 3571, 233, 4211, 547,
Found in Database : 7, 11, 13, 19, 23, 31, 37, 59, 61, 79, 83, 89, 97, 103, 107, 127, 139, 149,