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Development of Algorithmic Constructions

14:21:13 20.Apr 2024

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Polynom = x^2+13x-37

1. Algorithm

a:=1;
b:=13;
c:=-37;


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+13x-37

f(0)=37
f(1)=23
f(2)=7
f(3)=11
f(4)=31
f(5)=53
f(6)=1
f(7)=103
f(8)=131
f(9)=1
f(10)=193
f(11)=227
f(12)=263
f(13)=43
f(14)=1
f(15)=383
f(16)=61
f(17)=1
f(18)=521
f(19)=571
f(20)=89
f(21)=677
f(22)=733
f(23)=113
f(24)=1
f(25)=83
f(26)=977
f(27)=149
f(28)=101
f(29)=1181
f(30)=179
f(31)=1327
f(32)=1
f(33)=1481
f(34)=223
f(35)=1
f(36)=157
f(37)=1
f(38)=1901
f(39)=181
f(40)=2083
f(41)=311
f(42)=2273
f(43)=2371
f(44)=353
f(45)=1
f(46)=2677
f(47)=1
f(48)=59
f(49)=3001
f(50)=283
f(51)=461
f(52)=3343
f(53)=3461
f(54)=3581
f(55)=1
f(56)=1
f(57)=67
f(58)=1
f(59)=4211
f(60)=1
f(61)=1
f(62)=659
f(63)=4751
f(64)=73
f(65)=719
f(66)=167
f(67)=5323
f(68)=5471
f(69)=1
f(70)=251
f(71)=5927
f(72)=79
f(73)=1
f(74)=173
f(75)=6563
f(76)=1
f(77)=1
f(78)=307
f(79)=1033
f(80)=673
f(81)=7577
f(82)=7753
f(83)=1
f(84)=8111
f(85)=8293
f(86)=1
f(87)=8663
f(88)=1
f(89)=9041
f(90)=1319
f(91)=857
f(92)=9623
f(93)=1
f(94)=911
f(95)=10223
f(96)=10427
f(97)=1
f(98)=293
f(99)=257

b) Substitution of the polynom
The polynom f(x)=x^2+13x-37  could be written as f(y)= y^2-79.25 with x=y-6.5

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+6.5
f'(x)>2x+12

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 : 37, 23, 7, 11, 31, 53, 1, 103, 131, 1, 193, 227, 263, 43, 1, 383, 61, 1, 521, 571,
Found in Database : 37, 23, 7, 11, 31, 53, 103, 131, 193, 227, 263, 43, 383, 61, 521, 571, 89, 677, 733, 113, 83, 977, 149, 101, 1181, 179, 1327, 1481, 223, 157, 1901, 181,
Found in Database : 7, 11, 23, 31, 37, 43, 53, 59, 61, 67, 73, 79, 83, 89, 101, 103, 113, 131, 149,