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Development of Algorithmic Constructions

01:07:00 20.Apr 2024

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Polynom = x^2+60x-797

1. Algorithm

a:=1;
b:=60;
c:=-797;


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+60x-797

f(0)=797
f(1)=23
f(2)=673
f(3)=19
f(4)=541
f(5)=59
f(6)=401
f(7)=41
f(8)=11
f(9)=1
f(10)=97
f(11)=1
f(12)=67
f(13)=1
f(14)=239
f(15)=1
f(16)=419
f(17)=1
f(18)=607
f(19)=1
f(20)=73
f(21)=113
f(22)=53
f(23)=139
f(24)=1
f(25)=83
f(26)=1439
f(27)=1
f(28)=1667
f(29)=223
f(30)=173
f(31)=1
f(32)=1
f(33)=71
f(34)=2399
f(35)=79
f(36)=2659
f(37)=349
f(38)=2927
f(39)=383
f(40)=3203
f(41)=1
f(42)=317
f(43)=227
f(44)=3779
f(45)=491
f(46)=4079
f(47)=1
f(48)=107
f(49)=1
f(50)=4703
f(51)=1
f(52)=457
f(53)=1
f(54)=233
f(55)=691
f(56)=1
f(57)=367
f(58)=6047
f(59)=389
f(60)=337
f(61)=823
f(62)=101
f(63)=1
f(64)=1
f(65)=229
f(66)=103
f(67)=241
f(68)=7907
f(69)=1013
f(70)=1
f(71)=1063
f(72)=8707
f(73)=557
f(74)=829
f(75)=1
f(76)=9539
f(77)=1
f(78)=9967
f(79)=1
f(80)=1
f(81)=1
f(82)=10847
f(83)=1
f(84)=11299
f(85)=131
f(86)=1069
f(87)=1499
f(88)=12227
f(89)=1
f(90)=12703
f(91)=809
f(92)=13187
f(93)=1
f(94)=13679
f(95)=1741
f(96)=1289
f(97)=1
f(98)=773
f(99)=467

b) Substitution of the polynom
The polynom f(x)=x^2+60x-797  could be written as f(y)= y^2-1697 with x=y-30

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+30
f'(x)>2x+59

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 : 797, 23, 673, 19, 541, 59, 401, 41, 11, 1, 97, 1, 67, 1, 239, 1, 419, 1, 607, 1,
Found in Database : 797, 23, 673, 19, 541, 59, 401, 41, 11, 97, 67, 239, 419, 607, 73, 113, 53, 139, 83, 1439, 1667, 223, 173, 71, 2399, 79, 2659, 349, 2927, 383,
Found in Database : 11, 19, 23, 41, 53, 59, 67, 71, 73, 79, 83, 97, 101, 103, 107, 113, 131, 139,