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Development of Algorithmic Constructions

09:31:17 19.Apr 2024

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Polynom = x^2+76x-269

1. Algorithm

a:=1;
b:=76;
c:=-269;


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+76x-269

f(0)=269
f(1)=3
f(2)=113
f(3)=1
f(4)=17
f(5)=1
f(6)=223
f(7)=13
f(8)=31
f(9)=1
f(10)=197
f(11)=43
f(12)=787
f(13)=37
f(14)=991
f(15)=137
f(16)=401
f(17)=41
f(18)=1423
f(19)=1
f(20)=127
f(21)=1
f(22)=1
f(23)=251
f(24)=2131
f(25)=47
f(26)=2383
f(27)=157
f(28)=881
f(29)=347
f(30)=71
f(31)=1
f(32)=3187
f(33)=1
f(34)=89
f(35)=1
f(36)=53
f(37)=163
f(38)=239
f(39)=1
f(40)=1
f(41)=283
f(42)=109
f(43)=101
f(44)=5011
f(45)=647
f(46)=1
f(47)=1
f(48)=5683
f(49)=61
f(50)=1
f(51)=97
f(52)=2129
f(53)=821
f(54)=1
f(55)=1
f(56)=419
f(57)=457
f(58)=1
f(59)=1
f(60)=607
f(61)=337
f(62)=8287
f(63)=1061
f(64)=2897
f(65)=139
f(66)=9103
f(67)=1
f(68)=107
f(69)=1217
f(70)=1
f(71)=1
f(72)=1
f(73)=1
f(74)=10831
f(75)=691
f(76)=3761
f(77)=1439
f(78)=11743
f(79)=499
f(80)=12211
f(81)=389
f(82)=4229
f(83)=1
f(84)=13171
f(85)=1
f(86)=1051
f(87)=1
f(88)=4721
f(89)=1
f(90)=863
f(91)=311
f(92)=15187
f(93)=1931
f(94)=5237
f(95)=1997
f(96)=439
f(97)=1
f(98)=1291
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+76x-269  could be written as f(y)= y^2-1713 with x=y-38

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+38
f'(x)>2x+75

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 : 269, 3, 113, 1, 17, 1, 223, 13, 31, 1, 197, 43, 787, 37, 991, 137, 401, 41, 1423, 1,
Found in Database : 269, 3, 113, 17, 223, 13, 31, 197, 43, 787, 37, 991, 137, 401, 41, 1423, 127, 251, 2131, 47, 2383, 157, 881, 347, 71, 3187, 89, 53, 163, 239,
Found in Database : 3, 13, 17, 31, 37, 41, 43, 47, 53, 61, 71, 89, 97, 101, 107, 109, 113, 127, 137, 139,