---

Development of Algorithmic Constructions

16:15:14 18.Apr 2024

---

Polynom = x^2-240x+263

1. Algorithm

a:=1;
b:=-240;
c:=263;


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^2-240x+263

f(0)=263
f(1)=3
f(2)=71
f(3)=7
f(4)=227
f(5)=19
f(6)=163
f(7)=1
f(8)=59
f(9)=1
f(10)=97
f(11)=47
f(12)=2473
f(13)=1
f(14)=967
f(15)=389
f(16)=41
f(17)=1
f(18)=3733
f(19)=1
f(20)=197
f(21)=271
f(22)=1511
f(23)=1
f(24)=37
f(25)=1
f(26)=31
f(27)=1
f(28)=61
f(29)=1
f(30)=6037
f(31)=1
f(32)=2131
f(33)=821
f(34)=107
f(35)=1
f(36)=73
f(37)=151
f(38)=353
f(39)=947
f(40)=2579
f(41)=1
f(42)=8053
f(43)=1
f(44)=929
f(45)=1
f(46)=2887
f(47)=367
f(48)=1279
f(49)=379
f(50)=3079
f(51)=293
f(52)=1
f(53)=67
f(54)=9781
f(55)=1
f(56)=3347
f(57)=1
f(58)=1
f(59)=1
f(60)=257
f(61)=1
f(62)=1
f(63)=1361
f(64)=193
f(65)=463
f(66)=229
f(67)=1
f(68)=103
f(69)=1
f(70)=431
f(71)=1
f(72)=11833
f(73)=1
f(74)=4007
f(75)=757
f(76)=83
f(77)=1
f(78)=12373
f(79)=173
f(80)=199
f(81)=1
f(82)=4231
f(83)=1
f(84)=12841
f(85)=269
f(86)=4327
f(87)=233
f(88)=1
f(89)=1
f(90)=1
f(91)=277
f(92)=4451
f(93)=419
f(94)=641
f(95)=563
f(96)=191
f(97)=1
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-240x+263  could be written as f(y)= y^2-14137 with x=y+120

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-120
f'(x)>2x-241  with x > 119

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 : 263, 3, 71, 7, 227, 19, 163, 1, 59, 1, 97, 47, 2473, 1, 967, 389, 41, 1, 3733, 1,
Found in Database : 263, 3, 71, 7, 227, 19, 163, 59, 97, 47, 2473, 967, 389, 41, 3733, 197, 271, 1511, 37, 31, 61, 6037, 2131, 821, 107, 73, 151, 353, 947,
Found in Database : 3, 7, 19, 31, 37, 41, 47, 59, 61, 67, 71, 73, 83, 97, 103, 107,