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Development of Algorithmic Constructions

09:19:34 20.Apr 2024

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Polynom = x^2-53x+13

1. Algorithm

a:=1;
b:=-53;
c:=13;


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^2-53x+13

f(0)=13
f(1)=3
f(2)=89
f(3)=137
f(4)=61
f(5)=227
f(6)=269
f(7)=103
f(8)=347
f(9)=383
f(10)=139
f(11)=449
f(12)=479
f(13)=1
f(14)=41
f(15)=557
f(16)=193
f(17)=599
f(18)=617
f(19)=211
f(20)=647
f(21)=659
f(22)=223
f(23)=677
f(24)=683
f(25)=229
f(26)=53
f(27)=1
f(28)=1
f(29)=1
f(30)=1
f(31)=1
f(32)=1
f(33)=1
f(34)=1
f(35)=1
f(36)=1
f(37)=1
f(38)=1
f(39)=1
f(40)=1
f(41)=1
f(42)=1
f(43)=1
f(44)=1
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)=67
f(55)=1
f(56)=181
f(57)=241
f(58)=101
f(59)=367
f(60)=433
f(61)=167
f(62)=571
f(63)=643
f(64)=239
f(65)=1
f(66)=1
f(67)=317
f(68)=1033
f(69)=1117
f(70)=401
f(71)=1291
f(72)=1381
f(73)=491
f(74)=1567
f(75)=1663
f(76)=587
f(77)=1861
f(78)=151
f(79)=1
f(80)=1
f(81)=2281
f(82)=797
f(83)=2503
f(84)=2617
f(85)=911
f(86)=2851
f(87)=2971
f(88)=1031
f(89)=3217
f(90)=3343
f(91)=1
f(92)=277
f(93)=3733
f(94)=1289
f(95)=4003
f(96)=1
f(97)=1427
f(98)=4423
f(99)=4567

b) Substitution of the polynom
The polynom f(x)=x^2-53x+13  could be written as f(y)= y^2-689.25 with x=y+26.5

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-26.5
f'(x)>2x-54  with x > 26

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 : 13, 3, 89, 137, 61, 227, 269, 103, 347, 383, 139, 449, 479, 1, 41, 557, 193, 599, 617, 211,
Found in Database : 13, 3, 89, 137, 61, 227, 269, 103, 347, 383, 139, 449, 479, 41, 557, 193, 599, 617, 211, 647, 659, 223, 677, 683, 229, 53,
Found in Database : 3, 13, 41, 53, 61, 67, 89, 101, 103, 137, 139,