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Development of Algorithmic Constructions

09:17:32 16.Apr 2024

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Polynom = x^2+252x-2309

1. Algorithm

a:=1;
b:=252;
c:=-2309;


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+252x-2309

f(0)=2309
f(1)=257
f(2)=1801
f(3)=193
f(4)=5
f(5)=1
f(6)=761
f(7)=31
f(8)=229
f(9)=1
f(10)=311
f(11)=73
f(12)=859
f(13)=71
f(14)=283
f(15)=53
f(16)=1979
f(17)=1
f(18)=2551
f(19)=1
f(20)=101
f(21)=107
f(22)=3719
f(23)=251
f(24)=863
f(25)=577
f(26)=4919
f(27)=653
f(28)=5531
f(29)=1
f(30)=6151
f(31)=1
f(32)=6779
f(33)=887
f(34)=1483
f(35)=967
f(36)=8059
f(37)=131
f(38)=281
f(39)=113
f(40)=9371
f(41)=1213
f(42)=10039
f(43)=1297
f(44)=2143
f(45)=691
f(46)=11399
f(47)=367
f(48)=1
f(49)=1
f(50)=12791
f(51)=1
f(52)=13499
f(53)=433
f(54)=2843
f(55)=911
f(56)=14939
f(57)=1913
f(58)=15671
f(59)=401
f(60)=16411
f(61)=1049
f(62)=17159
f(63)=137
f(64)=3583
f(65)=2287
f(66)=18679
f(67)=2383
f(68)=1
f(69)=1
f(70)=20231
f(71)=1289
f(72)=21019
f(73)=2677
f(74)=4363
f(75)=2777
f(76)=22619
f(77)=1439
f(78)=23431
f(79)=149
f(80)=24251
f(81)=3083
f(82)=809
f(83)=3187
f(84)=1
f(85)=823
f(86)=26759
f(87)=1699
f(88)=27611
f(89)=701
f(90)=1
f(91)=3613
f(92)=29339
f(93)=1861
f(94)=6043
f(95)=479
f(96)=227
f(97)=3943
f(98)=31991
f(99)=811

b) Substitution of the polynom
The polynom f(x)=x^2+252x-2309  could be written as f(y)= y^2-18185 with x=y-126

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+126
f'(x)>2x+251

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 : 2309, 257, 1801, 193, 5, 1, 761, 31, 229, 1, 311, 73, 859, 71, 283, 53, 1979, 1, 2551, 1,
Found in Database : 2309, 257, 1801, 193, 5, 761, 31, 229, 311, 73, 859, 71, 283, 53, 1979, 2551, 101, 107, 3719, 251, 863, 577, 4919, 653, 5531, 6151, 6779, 887, 1483, 967, 8059, 131, 281, 113,
Found in Database : 5, 31, 53, 71, 73, 101, 107, 113, 131, 137, 149,