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Development of Algorithmic Constructions

08:05:01 19.Apr 2024

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Polynom = x^2+48x+7

1. Algorithm

a:=1;
b:=48;
c:=7;


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+48x+7

f(0)=7
f(1)=1
f(2)=107
f(3)=5
f(4)=43
f(5)=17
f(6)=331
f(7)=1
f(8)=13
f(9)=1
f(10)=587
f(11)=41
f(12)=727
f(13)=1
f(14)=1
f(15)=1
f(16)=1031
f(17)=139
f(18)=239
f(19)=1
f(20)=1367
f(21)=1
f(22)=1
f(23)=1
f(24)=347
f(25)=229
f(26)=1931
f(27)=127
f(28)=61
f(29)=1
f(30)=2347
f(31)=307
f(32)=151
f(33)=67
f(34)=1
f(35)=1
f(36)=433
f(37)=197
f(38)=131
f(39)=1
f(40)=3527
f(41)=457
f(42)=541
f(43)=1
f(44)=811
f(45)=1
f(46)=71
f(47)=1
f(48)=1
f(49)=1
f(50)=701
f(51)=79
f(52)=1
f(53)=1
f(54)=1103
f(55)=709
f(56)=1
f(57)=1
f(58)=1231
f(59)=1
f(60)=499
f(61)=1
f(62)=6827
f(63)=1
f(64)=1
f(65)=919
f(66)=443
f(67)=241
f(68)=1579
f(69)=101
f(70)=1181
f(71)=1
f(72)=8647
f(73)=1
f(74)=1
f(75)=577
f(76)=9431
f(77)=1
f(78)=281
f(79)=251
f(80)=10247
f(81)=1307
f(82)=10667
f(83)=1
f(84)=317
f(85)=1
f(86)=887
f(87)=113
f(88)=479
f(89)=1
f(90)=1
f(91)=1
f(92)=263
f(93)=1
f(94)=2671
f(95)=1699
f(96)=13831
f(97)=1759
f(98)=409
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+48x+7  could be written as f(y)= y^2-569 with x=y-24

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+24
f'(x)>2x+47

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 : 7, 1, 107, 5, 43, 17, 331, 1, 13, 1, 587, 41, 727, 1, 1, 1, 1031, 139, 239, 1,
Found in Database : 7, 107, 5, 43, 17, 331, 13, 587, 41, 727, 1031, 139, 239, 1367, 347, 229, 1931, 127, 61, 2347, 307, 151, 67, 433, 197, 131,
Found in Database : 5, 7, 13, 17, 41, 43, 61, 67, 71, 79, 101, 107, 113, 127, 131, 139,