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Development of Algorithmic Constructions

09:39:59 19.Apr 2024

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Polynom = x^2+80x-41

1. Algorithm

a:=1;
b:=80;
c:=-41;


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+80x-41

f(0)=41
f(1)=5
f(2)=3
f(3)=13
f(4)=59
f(5)=1
f(6)=19
f(7)=71
f(8)=17
f(9)=1
f(10)=859
f(11)=1
f(12)=1063
f(13)=73
f(14)=1
f(15)=173
f(16)=23
f(17)=67
f(18)=1723
f(19)=1
f(20)=653
f(21)=1
f(22)=2203
f(23)=97
f(24)=491
f(25)=1
f(26)=181
f(27)=89
f(28)=157
f(29)=1
f(30)=3259
f(31)=1
f(32)=1181
f(33)=461
f(34)=1
f(35)=83
f(36)=827
f(37)=1
f(38)=1481
f(39)=1
f(40)=4759
f(41)=1
f(42)=1
f(43)=1
f(44)=1
f(45)=349
f(46)=1151
f(47)=1
f(48)=359
f(49)=1
f(50)=2153
f(51)=1
f(52)=6823
f(53)=1
f(54)=1439
f(55)=1
f(56)=101
f(57)=971
f(58)=7963
f(59)=1
f(60)=643
f(61)=107
f(62)=127
f(63)=1
f(64)=367
f(65)=1
f(66)=1
f(67)=613
f(68)=257
f(69)=1
f(70)=10459
f(71)=1
f(72)=10903
f(73)=1
f(74)=757
f(75)=1
f(76)=139
f(77)=251
f(78)=1
f(79)=313
f(80)=4253
f(81)=1
f(82)=1
f(83)=281
f(84)=1
f(85)=1
f(86)=1
f(87)=1811
f(88)=641
f(89)=1
f(90)=15259
f(91)=1
f(92)=5261
f(93)=1
f(94)=1
f(95)=691
f(96)=3371
f(97)=2141
f(98)=5801
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+80x-41  could be written as f(y)= y^2-1641 with x=y-40

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+40
f'(x)>2x+79

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 : 41, 5, 3, 13, 59, 1, 19, 71, 17, 1, 859, 1, 1063, 73, 1, 173, 23, 67, 1723, 1,
Found in Database : 41, 5, 3, 13, 59, 19, 71, 17, 859, 1063, 73, 173, 23, 67, 1723, 653, 2203, 97, 491, 181, 89, 157, 3259, 1181, 461, 83, 827, 1481,
Found in Database : 3, 5, 13, 17, 19, 23, 41, 59, 67, 71, 73, 83, 89, 97, 101, 107, 127, 139,