---

Development of Algorithmic Constructions

02:31:36 18.Apr 2024

---

Polynom = x^2-88x+53

1. Algorithm

a:=1;
b:=-88;
c:=53;


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^2-88x+53

f(0)=53
f(1)=17
f(2)=7
f(3)=101
f(4)=283
f(5)=181
f(6)=439
f(7)=257
f(8)=587
f(9)=47
f(10)=727
f(11)=397
f(12)=859
f(13)=461
f(14)=983
f(15)=521
f(16)=157
f(17)=577
f(18)=71
f(19)=37
f(20)=1307
f(21)=677
f(22)=1399
f(23)=103
f(24)=1483
f(25)=761
f(26)=1559
f(27)=797
f(28)=1627
f(29)=829
f(30)=241
f(31)=857
f(32)=1
f(33)=881
f(34)=1783
f(35)=1
f(36)=107
f(37)=131
f(38)=1847
f(39)=929
f(40)=1867
f(41)=937
f(42)=1879
f(43)=941
f(44)=269
f(45)=1
f(46)=1
f(47)=1
f(48)=1
f(49)=1
f(50)=1
f(51)=1
f(52)=1
f(53)=1
f(54)=1
f(55)=1
f(56)=1
f(57)=1
f(58)=1
f(59)=1
f(60)=1
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)=233
f(91)=163
f(92)=421
f(93)=1
f(94)=617
f(95)=359
f(96)=821
f(97)=463
f(98)=1033
f(99)=571

b) Substitution of the polynom
The polynom f(x)=x^2-88x+53  could be written as f(y)= y^2-1883 with x=y+44

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-44
f'(x)>2x-89  with x > 43

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 : 53, 17, 7, 101, 283, 181, 439, 257, 587, 47, 727, 397, 859, 461, 983, 521, 157, 577, 71, 37,
Found in Database : 53, 17, 7, 101, 283, 181, 439, 257, 587, 47, 727, 397, 859, 461, 983, 521, 157, 577, 71, 37, 1307, 677, 1399, 103, 1483, 761, 1559, 797, 1627, 829, 241, 857, 881, 1783, 107, 131, 1847, 929,
Found in Database : 7, 17, 37, 47, 53, 71, 101, 103, 107, 131, 149,