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Development of Algorithmic Constructions

06:26:43 20.Apr 2024

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Polynom = x^2+23x-97

1. Algorithm

a:=1;
b:=23;
c:=-97;


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+23x-97

f(0)=97
f(1)=73
f(2)=47
f(3)=19
f(4)=11
f(5)=43
f(6)=7
f(7)=113
f(8)=151
f(9)=191
f(10)=233
f(11)=277
f(12)=17
f(13)=53
f(14)=421
f(15)=1
f(16)=31
f(17)=1
f(18)=641
f(19)=701
f(20)=109
f(21)=827
f(22)=1
f(23)=1
f(24)=1031
f(25)=1103
f(26)=107
f(27)=179
f(28)=1
f(29)=83
f(30)=1493
f(31)=1
f(32)=1663
f(33)=103
f(34)=263
f(35)=1933
f(36)=2027
f(37)=193
f(38)=2221
f(39)=211
f(40)=2423
f(41)=1
f(42)=2633
f(43)=2741
f(44)=2851
f(45)=2963
f(46)=181
f(47)=1
f(48)=1
f(49)=1
f(50)=1
f(51)=3677
f(52)=3803
f(53)=3931
f(54)=131
f(55)=599
f(56)=4327
f(57)=4463
f(58)=1
f(59)=431
f(60)=257
f(61)=457
f(62)=739
f(63)=313
f(64)=5471
f(65)=5623
f(66)=1
f(67)=349
f(68)=6091
f(69)=1
f(70)=1
f(71)=6577
f(72)=613
f(73)=6911
f(74)=1
f(75)=7253
f(76)=1061
f(77)=7603
f(78)=251
f(79)=419
f(80)=479
f(81)=757
f(82)=8513
f(83)=1
f(84)=523
f(85)=293
f(86)=9277
f(87)=9473
f(88)=509
f(89)=9871
f(90)=1439
f(91)=239
f(92)=953
f(93)=10691
f(94)=991
f(95)=11113
f(96)=241
f(97)=1
f(98)=619
f(99)=11981

b) Substitution of the polynom
The polynom f(x)=x^2+23x-97  could be written as f(y)= y^2-229.25 with x=y-11.5

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+11.5
f'(x)>2x+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 : 97, 73, 47, 19, 11, 43, 7, 113, 151, 191, 233, 277, 17, 53, 421, 1, 31, 1, 641, 701,
Found in Database : 97, 73, 47, 19, 11, 43, 7, 113, 151, 191, 233, 277, 17, 53, 421, 31, 641, 701, 109, 827, 1031, 1103, 107, 179, 83, 1493, 1663, 103, 263, 1933, 2027, 193, 2221, 211,
Found in Database : 7, 11, 17, 19, 31, 43, 47, 53, 73, 83, 97, 103, 107, 109, 113, 131, 139,