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Development of Algorithmic Constructions

11:36:02 19.Apr 2024

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Polynom = x^2+20x-31

1. Algorithm

a:=1;
b:=20;
c:=-31;


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+20x-31

f(0)=31
f(1)=5
f(2)=13
f(3)=19
f(4)=1
f(5)=47
f(6)=1
f(7)=79
f(8)=193
f(9)=23
f(10)=269
f(11)=1
f(12)=353
f(13)=199
f(14)=89
f(15)=1
f(16)=109
f(17)=1
f(18)=653
f(19)=71
f(20)=769
f(21)=83
f(22)=1
f(23)=479
f(24)=41
f(25)=547
f(26)=233
f(27)=619
f(28)=101
f(29)=139
f(30)=113
f(31)=1
f(32)=1
f(33)=859
f(34)=1
f(35)=947
f(36)=397
f(37)=1039
f(38)=53
f(39)=227
f(40)=103
f(41)=1
f(42)=1
f(43)=1
f(44)=557
f(45)=1447
f(46)=601
f(47)=1559
f(48)=61
f(49)=67
f(50)=3469
f(51)=359
f(52)=1
f(53)=1
f(54)=1
f(55)=1
f(56)=1
f(57)=2179
f(58)=4493
f(59)=463
f(60)=251
f(61)=491
f(62)=163
f(63)=1
f(64)=1069
f(65)=1
f(66)=1129
f(67)=223
f(68)=5953
f(69)=1
f(70)=6269
f(71)=643
f(72)=347
f(73)=1
f(74)=277
f(75)=3547
f(76)=1453
f(77)=3719
f(78)=331
f(79)=1
f(80)=613
f(81)=1
f(82)=641
f(83)=4259
f(84)=1741
f(85)=4447
f(86)=1
f(87)=4639
f(88)=9473
f(89)=967
f(90)=1
f(91)=1
f(92)=10273
f(93)=1
f(94)=2137
f(95)=419
f(96)=2221
f(97)=5659
f(98)=607
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+20x-31  could be written as f(y)= y^2-131 with x=y-10

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+10
f'(x)>2x+19

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 : 31, 5, 13, 19, 1, 47, 1, 79, 193, 23, 269, 1, 353, 199, 89, 1, 109, 1, 653, 71,
Found in Database : 31, 5, 13, 19, 47, 79, 193, 23, 269, 353, 199, 89, 109, 653, 71, 769, 83, 479, 41, 547, 233, 619, 101, 139, 113, 859, 947, 397, 1039, 53, 227,
Found in Database : 5, 13, 19, 23, 31, 41, 47, 53, 61, 67, 71, 79, 83, 89, 101, 103, 109, 113, 127, 131, 139,