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Development of Algorithmic Constructions

00:43:27 19.Apr 2024

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Polynom = x^2+92x-997

1. Algorithm

a:=1;
b:=92;
c:=-997;


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+92x-997

f(0)=997
f(1)=113
f(2)=809
f(3)=89
f(4)=613
f(5)=1
f(6)=409
f(7)=19
f(8)=197
f(9)=11
f(10)=23
f(11)=17
f(12)=251
f(13)=1
f(14)=487
f(15)=1
f(16)=43
f(17)=107
f(18)=983
f(19)=139
f(20)=1
f(21)=1
f(22)=1511
f(23)=103
f(24)=1787
f(25)=241
f(26)=109
f(27)=277
f(28)=1
f(29)=157
f(30)=2663
f(31)=1
f(32)=2971
f(33)=1
f(34)=173
f(35)=431
f(36)=1
f(37)=59
f(38)=3943
f(39)=257
f(40)=4283
f(41)=557
f(42)=421
f(43)=601
f(44)=4987
f(45)=1
f(46)=5351
f(47)=1
f(48)=97
f(49)=739
f(50)=359
f(51)=787
f(52)=6491
f(53)=1
f(54)=71
f(55)=443
f(56)=317
f(57)=937
f(58)=7703
f(59)=1
f(60)=8123
f(61)=521
f(62)=503
f(63)=137
f(64)=1
f(65)=1151
f(66)=9431
f(67)=1
f(68)=9883
f(69)=79
f(70)=10343
f(71)=661
f(72)=569
f(73)=1381
f(74)=11287
f(75)=131
f(76)=149
f(77)=751
f(78)=12263
f(79)=1
f(80)=12763
f(81)=1627
f(82)=577
f(83)=1
f(84)=811
f(85)=439
f(86)=1301
f(87)=911
f(88)=14843
f(89)=1889
f(90)=15383
f(91)=1
f(92)=179
f(93)=1013
f(94)=16487
f(95)=1
f(96)=1
f(97)=1
f(98)=17623
f(99)=2239

b) Substitution of the polynom
The polynom f(x)=x^2+92x-997  could be written as f(y)= y^2-3113 with x=y-46

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+46
f'(x)>2x+91

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 : 997, 113, 809, 89, 613, 1, 409, 19, 197, 11, 23, 17, 251, 1, 487, 1, 43, 107, 983, 139,
Found in Database : 997, 113, 809, 89, 613, 409, 19, 197, 11, 23, 17, 251, 487, 43, 107, 983, 139, 1511, 103, 1787, 241, 109, 277, 157, 2663, 2971, 173, 431, 59, 3943, 257,
Found in Database : 11, 17, 19, 23, 43, 59, 71, 79, 89, 97, 103, 107, 109, 113, 131, 137, 139, 149,