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Development of Algorithmic Constructions

15:33:39 20.Apr 2024

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Polynom = x^2+212x-2557

1. Algorithm

a:=1;
b:=212;
c:=-2557;


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+212x-2557

f(0)=2557
f(1)=293
f(2)=2129
f(3)=239
f(4)=1693
f(5)=23
f(6)=1249
f(7)=1
f(8)=797
f(9)=71
f(10)=337
f(11)=13
f(12)=131
f(13)=1
f(14)=607
f(15)=53
f(16)=1091
f(17)=167
f(18)=1583
f(19)=229
f(20)=2083
f(21)=73
f(22)=2591
f(23)=89
f(24)=1
f(25)=421
f(26)=3631
f(27)=487
f(28)=181
f(29)=277
f(30)=4703
f(31)=311
f(32)=59
f(33)=691
f(34)=5807
f(35)=761
f(36)=1
f(37)=1
f(38)=1
f(39)=113
f(40)=7523
f(41)=977
f(42)=8111
f(43)=1051
f(44)=8707
f(45)=563
f(46)=9311
f(47)=601
f(48)=9923
f(49)=1279
f(50)=811
f(51)=1
f(52)=11171
f(53)=359
f(54)=11807
f(55)=379
f(56)=12451
f(57)=1597
f(58)=13103
f(59)=1
f(60)=13763
f(61)=881
f(62)=14431
f(63)=1
f(64)=15107
f(65)=1931
f(66)=15791
f(67)=2017
f(68)=1
f(69)=263
f(70)=17183
f(71)=137
f(72)=17891
f(73)=2281
f(74)=809
f(75)=2371
f(76)=1487
f(77)=1231
f(78)=20063
f(79)=1277
f(80)=1
f(81)=2647
f(82)=937
f(83)=2741
f(84)=22307
f(85)=709
f(86)=23071
f(87)=733
f(88)=211
f(89)=233
f(90)=24623
f(91)=1
f(92)=25411
f(93)=1613
f(94)=1
f(95)=1663
f(96)=27011
f(97)=149
f(98)=27823
f(99)=3529

b) Substitution of the polynom
The polynom f(x)=x^2+212x-2557  could be written as f(y)= y^2-13793 with x=y-106

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+106
f'(x)>2x+211

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 : 2557, 293, 2129, 239, 1693, 23, 1249, 1, 797, 71, 337, 13, 131, 1, 607, 53, 1091, 167, 1583, 229,
Found in Database : 2557, 293, 2129, 239, 1693, 23, 1249, 797, 71, 337, 13, 131, 607, 53, 1091, 167, 1583, 229, 2083, 73, 2591, 89, 421, 3631, 487, 181, 277, 4703, 311, 59, 691, 5807, 761, 113,
Found in Database : 13, 23, 53, 59, 71, 73, 89, 113, 131, 137, 149,