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Development of Algorithmic Constructions

04:56:00 20.Apr 2024

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Polynom = x^2+60x-13

1. Algorithm

a:=1;
b:=60;
c:=-13;


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+60x-13

f(0)=13
f(1)=3
f(2)=37
f(3)=11
f(4)=1
f(5)=1
f(6)=383
f(7)=19
f(8)=59
f(9)=1
f(10)=229
f(11)=1
f(12)=23
f(13)=1
f(14)=31
f(15)=139
f(16)=401
f(17)=1
f(18)=107
f(19)=1
f(20)=1
f(21)=211
f(22)=199
f(23)=79
f(24)=2003
f(25)=1
f(26)=1
f(27)=73
f(28)=43
f(29)=1
f(30)=2687
f(31)=1
f(32)=977
f(33)=191
f(34)=1061
f(35)=1
f(36)=313
f(37)=149
f(38)=1237
f(39)=1
f(40)=443
f(41)=1
f(42)=4271
f(43)=1
f(44)=1
f(45)=1
f(46)=1621
f(47)=1
f(48)=5171
f(49)=1
f(50)=1
f(51)=353
f(52)=1
f(53)=83
f(54)=6143
f(55)=263
f(56)=2161
f(57)=1
f(58)=1
f(59)=1
f(60)=7187
f(61)=307
f(62)=839
f(63)=967
f(64)=1
f(65)=1
f(66)=1
f(67)=1
f(68)=2897
f(69)=101
f(70)=233
f(71)=1
f(72)=9491
f(73)=1
f(74)=3301
f(75)=1
f(76)=1
f(77)=439
f(78)=827
f(79)=457
f(80)=113
f(81)=1
f(82)=3877
f(83)=1
f(84)=281
f(85)=1
f(86)=1
f(87)=1597
f(88)=4337
f(89)=1
f(90)=13487
f(91)=1
f(92)=4657
f(93)=1777
f(94)=1607
f(95)=613
f(96)=1151
f(97)=317
f(98)=1
f(99)=983

b) Substitution of the polynom
The polynom f(x)=x^2+60x-13  could be written as f(y)= y^2-913 with x=y-30

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+30
f'(x)>2x+59

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 : 13, 3, 37, 11, 1, 1, 383, 19, 59, 1, 229, 1, 23, 1, 31, 139, 401, 1, 107, 1,
Found in Database : 13, 3, 37, 11, 383, 19, 59, 229, 23, 31, 139, 401, 107, 211, 199, 79, 2003, 73, 43, 2687, 977, 191, 1061, 313, 149, 1237,
Found in Database : 3, 11, 13, 19, 23, 31, 37, 43, 59, 73, 79, 83, 101, 107, 113, 139, 149,