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Development of Algorithmic Constructions

14:01:14 20.Apr 2024

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Polynom = x^2-238x-3

1. Algorithm

a:=1;
b:=-238;
c:=-3;


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^2-238x-3

f(0)=3
f(1)=5
f(2)=19
f(3)=59
f(4)=313
f(5)=73
f(6)=31
f(7)=1
f(8)=97
f(9)=43
f(10)=761
f(11)=1
f(12)=181
f(13)=61
f(14)=1
f(15)=1
f(16)=79
f(17)=47
f(18)=1321
f(19)=347
f(20)=4363
f(21)=1
f(22)=317
f(23)=1237
f(24)=571
f(25)=37
f(26)=1103
f(27)=1
f(28)=53
f(29)=379
f(30)=2081
f(31)=107
f(32)=1319
f(33)=1
f(34)=257
f(35)=1777
f(36)=1
f(37)=1
f(38)=7603
f(39)=647
f(40)=139
f(41)=101
f(42)=1
f(43)=233
f(44)=8539
f(45)=1
f(46)=1
f(47)=449
f(48)=3041
f(49)=193
f(50)=9403
f(51)=1
f(52)=1
f(53)=613
f(54)=3313
f(55)=839
f(56)=2039
f(57)=1
f(58)=1
f(59)=1
f(60)=1187
f(61)=1
f(62)=1
f(63)=919
f(64)=1
f(65)=1
f(66)=757
f(67)=191
f(68)=373
f(69)=1
f(70)=1307
f(71)=593
f(72)=797
f(73)=251
f(74)=199
f(75)=1019
f(76)=821
f(77)=1
f(78)=1
f(79)=349
f(80)=269
f(81)=1
f(82)=853
f(83)=3217
f(84)=227
f(85)=271
f(86)=523
f(87)=1
f(88)=163
f(89)=829
f(90)=4441
f(91)=223
f(92)=2687
f(93)=281
f(94)=4513
f(95)=1
f(96)=1
f(97)=1
f(98)=13723
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-238x-3  could be written as f(y)= y^2-14164 with x=y+119

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-119
f'(x)>2x-239  with x > 119

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 : 3, 5, 19, 59, 313, 73, 31, 1, 97, 43, 761, 1, 181, 61, 1, 1, 79, 47, 1321, 347,
Found in Database : 3, 5, 19, 59, 313, 73, 31, 97, 43, 761, 181, 61, 79, 47, 1321, 347, 4363, 317, 1237, 571, 37, 1103, 53, 379, 2081, 107, 1319, 257, 1777, 7603, 647,
Found in Database : 3, 5, 19, 31, 37, 43, 47, 53, 59, 61, 73, 79, 97, 101, 107, 131, 139, 149,