---

Development of Algorithmic Constructions

07:05:54 19.Apr 2024

---

Polynom = x^2+48x-241

1. Algorithm

a:=1;
b:=48;
c:=-241;


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-241

f(0)=241
f(1)=3
f(2)=47
f(3)=11
f(4)=1
f(5)=1
f(6)=83
f(7)=1
f(8)=23
f(9)=17
f(10)=113
f(11)=1
f(12)=479
f(13)=1
f(14)=19
f(15)=1
f(16)=29
f(17)=1
f(18)=947
f(19)=43
f(20)=373
f(21)=151
f(22)=433
f(23)=1
f(24)=1487
f(25)=1
f(26)=1
f(27)=223
f(28)=37
f(29)=1
f(30)=2099
f(31)=1
f(32)=773
f(33)=1
f(34)=283
f(35)=1
f(36)=1
f(37)=1
f(38)=1009
f(39)=197
f(40)=1093
f(41)=71
f(42)=3539
f(43)=1
f(44)=1
f(45)=1
f(46)=1361
f(47)=1
f(48)=397
f(49)=1
f(50)=1553
f(51)=601
f(52)=1
f(53)=1
f(54)=229
f(55)=1
f(56)=1861
f(57)=359
f(58)=179
f(59)=1
f(60)=367
f(61)=89
f(62)=1
f(63)=211
f(64)=2309
f(65)=1
f(66)=7283
f(67)=311
f(68)=2549
f(69)=1
f(70)=1
f(71)=1
f(72)=227
f(73)=1
f(74)=101
f(75)=1123
f(76)=3061
f(77)=1
f(78)=9587
f(79)=1
f(80)=1
f(81)=1
f(82)=1
f(83)=443
f(84)=10847
f(85)=461
f(86)=3761
f(87)=719
f(88)=1303
f(89)=1
f(90)=641
f(91)=1
f(92)=383
f(93)=1609
f(94)=257
f(95)=139
f(96)=1
f(97)=1
f(98)=521
f(99)=1789

b) Substitution of the polynom
The polynom f(x)=x^2+48x-241  could be written as f(y)= y^2-817 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 : 241, 3, 47, 11, 1, 1, 83, 1, 23, 17, 113, 1, 479, 1, 19, 1, 29, 1, 947, 43,
Found in Database : 241, 3, 47, 11, 83, 23, 17, 113, 479, 19, 29, 947, 43, 373, 151, 433, 1487, 223, 37, 2099, 773, 283, 1009, 197,
Found in Database : 3, 11, 17, 19, 23, 29, 37, 43, 47, 71, 83, 89, 101, 113, 139,