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Development of Algorithmic Constructions

23:59:19 19.Apr 2024

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Polynom = x^2+36x-1013

1. Algorithm

a:=1;
b:=36;
c:=-1013;


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+36x-1013

f(0)=1013
f(1)=61
f(2)=937
f(3)=7
f(4)=853
f(5)=101
f(6)=761
f(7)=89
f(8)=661
f(9)=19
f(10)=79
f(11)=31
f(12)=23
f(13)=47
f(14)=313
f(15)=1
f(16)=181
f(17)=1
f(18)=41
f(19)=1
f(20)=107
f(21)=1
f(22)=263
f(23)=43
f(24)=1
f(25)=1
f(26)=599
f(27)=1
f(28)=1
f(29)=109
f(30)=967
f(31)=1
f(32)=1163
f(33)=1
f(34)=1367
f(35)=1
f(36)=1579
f(37)=211
f(38)=257
f(39)=239
f(40)=2027
f(41)=67
f(42)=73
f(43)=149
f(44)=1
f(45)=1
f(46)=1
f(47)=1
f(48)=3019
f(49)=197
f(50)=173
f(51)=1
f(52)=509
f(53)=463
f(54)=3847
f(55)=499
f(56)=4139
f(57)=1
f(58)=193
f(59)=1
f(60)=1
f(61)=613
f(62)=83
f(63)=653
f(64)=5387
f(65)=347
f(66)=1
f(67)=1
f(68)=1
f(69)=1
f(70)=1
f(71)=823
f(72)=6763
f(73)=1
f(74)=7127
f(75)=457
f(76)=7499
f(77)=1
f(78)=7879
f(79)=1009
f(80)=1181
f(81)=1
f(82)=8663
f(83)=277
f(84)=9067
f(85)=1
f(86)=9479
f(87)=1
f(88)=521
f(89)=1
f(90)=449
f(91)=659
f(92)=229
f(93)=1373
f(94)=1601
f(95)=1429
f(96)=131
f(97)=743
f(98)=12119
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+36x-1013  could be written as f(y)= y^2-1337 with x=y-18

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+18
f'(x)>2x+35

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 : 1013, 61, 937, 7, 853, 101, 761, 89, 661, 19, 79, 31, 23, 47, 313, 1, 181, 1, 41, 1,
Found in Database : 1013, 61, 937, 7, 853, 101, 761, 89, 661, 19, 79, 31, 23, 47, 313, 181, 41, 107, 263, 43, 599, 109, 967, 1163, 1367, 1579, 211, 257, 239,
Found in Database : 7, 19, 23, 31, 41, 43, 47, 61, 67, 73, 79, 83, 89, 101, 107, 109, 131, 139, 149,