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Development of Algorithmic Constructions

01:51:13 19.Apr 2024

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Polynom = x^2-232x+479

1. Algorithm

a:=1;
b:=-232;
c:=479;


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^2-232x+479

f(0)=479
f(1)=31
f(2)=19
f(3)=13
f(4)=433
f(5)=41
f(6)=877
f(7)=137
f(8)=101
f(9)=191
f(10)=1741
f(11)=61
f(12)=2161
f(13)=37
f(14)=83
f(15)=347
f(16)=229
f(17)=397
f(18)=3373
f(19)=223
f(20)=3761
f(21)=1
f(22)=1
f(23)=541
f(24)=4513
f(25)=587
f(26)=4877
f(27)=79
f(28)=5233
f(29)=1
f(30)=5581
f(31)=719
f(32)=1
f(33)=761
f(34)=1
f(35)=401
f(36)=6577
f(37)=421
f(38)=113
f(39)=881
f(40)=379
f(41)=919
f(42)=577
f(43)=239
f(44)=7793
f(45)=1
f(46)=197
f(47)=1
f(48)=8353
f(49)=1061
f(50)=233
f(51)=547
f(52)=107
f(53)=563
f(54)=9133
f(55)=89
f(56)=9377
f(57)=1187
f(58)=9613
f(59)=1
f(60)=757
f(61)=311
f(62)=10061
f(63)=1
f(64)=10273
f(65)=1297
f(66)=10477
f(67)=661
f(68)=821
f(69)=673
f(70)=10861
f(71)=1
f(72)=181
f(73)=1
f(74)=11213
f(75)=353
f(76)=367
f(77)=179
f(78)=607
f(79)=1451
f(80)=11681
f(81)=1
f(82)=11821
f(83)=743
f(84)=11953
f(85)=751
f(86)=929
f(87)=1
f(88)=1
f(89)=1531
f(90)=12301
f(91)=193
f(92)=12401
f(93)=389
f(94)=1
f(95)=1567
f(96)=12577
f(97)=1
f(98)=12653
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-232x+479  could be written as f(y)= y^2-12977 with x=y+116

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-116
f'(x)>2x-233  with x > 114

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 : 479, 31, 19, 13, 433, 41, 877, 137, 101, 191, 1741, 61, 2161, 37, 83, 347, 229, 397, 3373, 223,
Found in Database : 479, 31, 19, 13, 433, 41, 877, 137, 101, 191, 1741, 61, 2161, 37, 83, 347, 229, 397, 3373, 223, 3761, 541, 4513, 587, 4877, 79, 5233, 5581, 719, 761, 401, 6577, 421, 113, 881,
Found in Database : 13, 19, 31, 37, 41, 61, 79, 83, 89, 101, 107, 113, 137,