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Development of Algorithmic Constructions

09:06:49 19.Apr 2024

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Polynom = x^2-46x+59

1. Algorithm

a:=1;
b:=-46;
c:=59;


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^2-46x+59

f(0)=59
f(1)=7
f(2)=29
f(3)=5
f(4)=109
f(5)=73
f(6)=181
f(7)=107
f(8)=1
f(9)=137
f(10)=43
f(11)=163
f(12)=349
f(13)=37
f(14)=389
f(15)=1
f(16)=421
f(17)=31
f(18)=89
f(19)=227
f(20)=461
f(21)=233
f(22)=67
f(23)=47
f(24)=1
f(25)=1
f(26)=1
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)=53
f(48)=1
f(49)=103
f(50)=1
f(51)=157
f(52)=1
f(53)=1
f(54)=491
f(55)=277
f(56)=619
f(57)=1
f(58)=151
f(59)=1
f(60)=1
f(61)=487
f(62)=1051
f(63)=113
f(64)=173
f(65)=647
f(66)=197
f(67)=733
f(68)=311
f(69)=823
f(70)=1
f(71)=131
f(72)=1931
f(73)=1
f(74)=2131
f(75)=1117
f(76)=2339
f(77)=1223
f(78)=1
f(79)=1
f(80)=397
f(81)=1447
f(82)=3011
f(83)=313
f(84)=3251
f(85)=241
f(86)=3499
f(87)=1
f(88)=751
f(89)=1
f(90)=4019
f(91)=1
f(92)=613
f(93)=443
f(94)=653
f(95)=2357
f(96)=1
f(97)=2503
f(98)=1031
f(99)=379

b) Substitution of the polynom
The polynom f(x)=x^2-46x+59  could be written as f(y)= y^2-470 with x=y+23

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-23
f'(x)>2x-47  with x > 22

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 : 59, 7, 29, 5, 109, 73, 181, 107, 1, 137, 43, 163, 349, 37, 389, 1, 421, 31, 89, 227,
Found in Database : 59, 7, 29, 5, 109, 73, 181, 107, 137, 43, 163, 349, 37, 389, 421, 31, 89, 227, 461, 233, 67, 47,
Found in Database : 5, 7, 29, 31, 37, 43, 47, 53, 59, 67, 73, 89, 103, 107, 109, 113, 131, 137,