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Development of Algorithmic Constructions

09:19:37 19.Apr 2024

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Polynom = x^2+40x-47

1. Algorithm

a:=1;
b:=40;
c:=-47;


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+40x-47

f(0)=47
f(1)=3
f(2)=37
f(3)=41
f(4)=43
f(5)=89
f(6)=229
f(7)=1
f(8)=337
f(9)=197
f(10)=151
f(11)=257
f(12)=577
f(13)=107
f(14)=709
f(15)=389
f(16)=283
f(17)=461
f(18)=997
f(19)=179
f(20)=1153
f(21)=617
f(22)=439
f(23)=701
f(24)=1489
f(25)=263
f(26)=1669
f(27)=881
f(28)=619
f(29)=977
f(30)=2053
f(31)=359
f(32)=61
f(33)=1181
f(34)=823
f(35)=1289
f(36)=2689
f(37)=467
f(38)=2917
f(39)=1
f(40)=1051
f(41)=1637
f(42)=79
f(43)=587
f(44)=1
f(45)=1889
f(46)=1303
f(47)=1
f(48)=4177
f(49)=719
f(50)=73
f(51)=2297
f(52)=1579
f(53)=2441
f(54)=1
f(55)=863
f(56)=1
f(57)=2741
f(58)=1879
f(59)=2897
f(60)=5953
f(61)=1019
f(62)=6277
f(63)=3221
f(64)=2203
f(65)=3389
f(66)=6949
f(67)=1187
f(68)=7297
f(69)=101
f(70)=2551
f(71)=3917
f(72)=8017
f(73)=1367
f(74)=8389
f(75)=4289
f(76)=1
f(77)=4481
f(78)=9157
f(79)=1559
f(80)=233
f(81)=4877
f(82)=3319
f(83)=5081
f(84)=10369
f(85)=1
f(86)=10789
f(87)=5501
f(88)=3739
f(89)=5717
f(90)=271
f(91)=1979
f(92)=12097
f(93)=1
f(94)=1
f(95)=6389
f(96)=13009
f(97)=2207
f(98)=13477
f(99)=6857

b) Substitution of the polynom
The polynom f(x)=x^2+40x-47  could be written as f(y)= y^2-447 with x=y-20

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+20
f'(x)>2x+39

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 : 47, 3, 37, 41, 43, 89, 229, 1, 337, 197, 151, 257, 577, 107, 709, 389, 283, 461, 997, 179,
Found in Database : 47, 3, 37, 41, 43, 89, 229, 337, 197, 151, 257, 577, 107, 709, 389, 283, 461, 997, 179, 1153, 617, 439, 701, 1489, 263, 1669, 881, 619, 977, 2053, 359, 61, 1181, 823, 1289, 2689, 467, 2917,
Found in Database : 3, 37, 41, 43, 47, 61, 73, 79, 89, 101, 107, 137, 139, 149,