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Development of Algorithmic Constructions

12:45:07 20.Apr 2024

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Polynom = x^2-120x-113

1. Algorithm

a:=1;
b:=-120;
c:=-113;


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^2-120x-113

f(0)=113
f(1)=29
f(2)=349
f(3)=1
f(4)=577
f(5)=43
f(6)=797
f(7)=1
f(8)=1009
f(9)=139
f(10)=1213
f(11)=41
f(12)=1409
f(13)=47
f(14)=1597
f(15)=211
f(16)=1777
f(17)=233
f(18)=1949
f(19)=127
f(20)=2113
f(21)=137
f(22)=2269
f(23)=293
f(24)=2417
f(25)=311
f(26)=2557
f(27)=1
f(28)=2689
f(29)=1
f(30)=97
f(31)=359
f(32)=101
f(33)=373
f(34)=3037
f(35)=193
f(36)=3137
f(37)=199
f(38)=3229
f(39)=409
f(40)=3313
f(41)=419
f(42)=3389
f(43)=107
f(44)=3457
f(45)=109
f(46)=3517
f(47)=443
f(48)=83
f(49)=449
f(50)=3613
f(51)=227
f(52)=89
f(53)=229
f(54)=3677
f(55)=461
f(56)=3697
f(57)=463
f(58)=3709
f(59)=1
f(60)=79
f(61)=1
f(62)=1
f(63)=1
f(64)=1
f(65)=1
f(66)=1
f(67)=1
f(68)=1
f(69)=1
f(70)=1
f(71)=1
f(72)=1
f(73)=1
f(74)=1
f(75)=1
f(76)=1
f(77)=1
f(78)=1
f(79)=1
f(80)=1
f(81)=1
f(82)=1
f(83)=1
f(84)=1
f(85)=1
f(86)=1
f(87)=1
f(88)=1
f(89)=1
f(90)=1
f(91)=1
f(92)=1
f(93)=1
f(94)=1
f(95)=1
f(96)=1
f(97)=1
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-120x-113  could be written as f(y)= y^2-3713 with x=y+60

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-60
f'(x)>2x-121  with x > 61

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 : 113, 29, 349, 1, 577, 43, 797, 1, 1009, 139, 1213, 41, 1409, 47, 1597, 211, 1777, 233, 1949, 127,
Found in Database : 113, 29, 349, 577, 43, 797, 1009, 139, 1213, 41, 1409, 47, 1597, 211, 1777, 233, 1949, 127, 2113, 137, 2269, 293, 2417, 311, 2557, 2689, 97, 359, 101, 373, 3037, 193, 3137, 199, 3229, 409,
Found in Database : 29, 41, 43, 47, 79, 83, 89, 97, 101, 107, 109, 113, 127, 131, 137, 139,