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Development of Algorithmic Constructions

21:17:11 19.Apr 2024

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Polynom = x^2+32x-307

1. Algorithm

a:=1;
b:=32;
c:=-307;


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+32x-307

f(0)=307
f(1)=137
f(2)=239
f(3)=101
f(4)=163
f(5)=61
f(6)=79
f(7)=17
f(8)=13
f(9)=31
f(10)=113
f(11)=83
f(12)=1
f(13)=139
f(14)=337
f(15)=199
f(16)=461
f(17)=263
f(18)=593
f(19)=331
f(20)=733
f(21)=1
f(22)=881
f(23)=479
f(24)=1
f(25)=43
f(26)=1201
f(27)=643
f(28)=1373
f(29)=1
f(30)=1553
f(31)=823
f(32)=1741
f(33)=919
f(34)=149
f(35)=1019
f(36)=2141
f(37)=1123
f(38)=181
f(39)=1231
f(40)=1
f(41)=1
f(42)=2801
f(43)=1459
f(44)=3037
f(45)=1579
f(46)=193
f(47)=131
f(48)=3533
f(49)=1831
f(50)=3793
f(51)=151
f(52)=1
f(53)=2099
f(54)=4337
f(55)=2239
f(56)=4621
f(57)=2383
f(58)=1
f(59)=2531
f(60)=401
f(61)=2683
f(62)=5521
f(63)=167
f(64)=449
f(65)=2999
f(66)=1
f(67)=3163
f(68)=1
f(69)=3331
f(70)=6833
f(71)=1
f(72)=1
f(73)=283
f(74)=7537
f(75)=227
f(76)=7901
f(77)=311
f(78)=8273
f(79)=4231
f(80)=509
f(81)=4423
f(82)=9041
f(83)=1
f(84)=9437
f(85)=1
f(86)=757
f(87)=5023
f(88)=10253
f(89)=5231
f(90)=821
f(91)=5443
f(92)=653
f(93)=5659
f(94)=1
f(95)=5879
f(96)=11981
f(97)=359
f(98)=12433
f(99)=487

b) Substitution of the polynom
The polynom f(x)=x^2+32x-307  could be written as f(y)= y^2-563 with x=y-16

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+16
f'(x)>2x+31

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 : 307, 137, 239, 101, 163, 61, 79, 17, 13, 31, 113, 83, 1, 139, 337, 199, 461, 263, 593, 331,
Found in Database : 307, 137, 239, 101, 163, 61, 79, 17, 13, 31, 113, 83, 139, 337, 199, 461, 263, 593, 331, 733, 881, 479, 43, 1201, 643, 1373, 1553, 823, 1741, 919, 149, 1019, 2141, 1123, 181, 1231,
Found in Database : 13, 17, 31, 43, 61, 79, 83, 101, 113, 131, 137, 139, 149,