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Development of Algorithmic Constructions

08:07:22 16.Apr 2024

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Polynom = x^2-160x+23

1. Algorithm

a:=1;
b:=-160;
c:=23;


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^2-160x+23

f(0)=23
f(1)=17
f(2)=293
f(3)=7
f(4)=601
f(5)=47
f(6)=53
f(7)=131
f(8)=1193
f(9)=167
f(10)=211
f(11)=101
f(12)=1753
f(13)=59
f(14)=43
f(15)=269
f(16)=2281
f(17)=1
f(18)=149
f(19)=83
f(20)=2777
f(21)=181
f(22)=1
f(23)=1
f(24)=463
f(25)=419
f(26)=3461
f(27)=223
f(28)=3673
f(29)=1
f(30)=3877
f(31)=71
f(32)=4073
f(33)=521
f(34)=4261
f(35)=1
f(36)=4441
f(37)=283
f(38)=659
f(39)=587
f(40)=281
f(41)=607
f(42)=4933
f(43)=313
f(44)=5081
f(45)=1
f(46)=227
f(47)=661
f(48)=1
f(49)=677
f(50)=5477
f(51)=173
f(52)=1
f(53)=353
f(54)=5701
f(55)=719
f(56)=5801
f(57)=1
f(58)=1
f(59)=1
f(60)=139
f(61)=1
f(62)=6053
f(63)=761
f(64)=6121
f(65)=769
f(66)=883
f(67)=97
f(68)=271
f(69)=1
f(70)=6277
f(71)=787
f(72)=107
f(73)=113
f(74)=373
f(75)=397
f(76)=6361
f(77)=199
f(78)=6373
f(79)=797
f(80)=911
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^2-160x+23  could be written as f(y)= y^2-6377 with x=y+80

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-80
f'(x)>2x-161  with x > 80

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 : 23, 17, 293, 7, 601, 47, 53, 131, 1193, 167, 211, 101, 1753, 59, 43, 269, 2281, 1, 149, 83,
Found in Database : 23, 17, 293, 7, 601, 47, 53, 131, 1193, 167, 211, 101, 1753, 59, 43, 269, 2281, 149, 83, 2777, 181, 463, 419, 3461, 223, 3673, 3877, 71, 4073, 521, 4261, 4441, 283, 659, 587,
Found in Database : 7, 17, 23, 43, 47, 53, 59, 71, 83, 97, 101, 107, 113, 131, 139, 149,