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Development of Algorithmic Constructions

23:35:12 19.Apr 2024

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Polynom = x^2+144x-337

1. Algorithm

a:=1;
b:=144;
c:=-337;


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+144x-337

f(0)=337
f(1)=3
f(2)=5
f(3)=13
f(4)=17
f(5)=1
f(6)=563
f(7)=1
f(8)=293
f(9)=1
f(10)=401
f(11)=19
f(12)=307
f(13)=71
f(14)=1
f(15)=1
f(16)=1
f(17)=1
f(18)=2579
f(19)=23
f(20)=109
f(21)=1
f(22)=1
f(23)=73
f(24)=739
f(25)=1
f(26)=1361
f(27)=107
f(28)=1493
f(29)=1
f(30)=257
f(31)=53
f(32)=353
f(33)=43
f(34)=127
f(35)=1
f(36)=6143
f(37)=1
f(38)=1
f(39)=1
f(40)=2341
f(41)=151
f(42)=1
f(43)=1
f(44)=1
f(45)=1021
f(46)=2801
f(47)=1
f(48)=683
f(49)=1
f(50)=3121
f(51)=1201
f(52)=1
f(53)=421
f(54)=1
f(55)=1
f(56)=1
f(57)=139
f(58)=3793
f(59)=97
f(60)=11903
f(61)=1
f(62)=829
f(63)=397
f(64)=173
f(65)=1
f(66)=13523
f(67)=1
f(68)=1
f(69)=359
f(70)=1627
f(71)=311
f(72)=179
f(73)=1
f(74)=1
f(75)=2011
f(76)=1
f(77)=1
f(78)=16979
f(79)=1
f(80)=5861
f(81)=1
f(82)=1213
f(83)=1
f(84)=1
f(85)=797
f(86)=6481
f(87)=1
f(88)=1
f(89)=1
f(90)=1
f(91)=877
f(92)=1
f(93)=2713
f(94)=113
f(95)=233
f(96)=1
f(97)=1
f(98)=7793
f(99)=593

b) Substitution of the polynom
The polynom f(x)=x^2+144x-337  could be written as f(y)= y^2-5521 with x=y-72

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+72
f'(x)>2x+143

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 : 337, 3, 5, 13, 17, 1, 563, 1, 293, 1, 401, 19, 307, 71, 1, 1, 1, 1, 2579, 23,
Found in Database : 337, 3, 5, 13, 17, 563, 293, 401, 19, 307, 71, 2579, 23, 109, 73, 739, 1361, 107, 1493, 257, 53, 353, 43, 127, 6143,
Found in Database : 3, 5, 13, 17, 19, 23, 43, 53, 71, 73, 97, 107, 109, 113, 127, 139,