---

Development of Algorithmic Constructions

05:33:49 19.Apr 2024

---

Polynom = x^2-38x+59

1. Algorithm

a:=1;
b:=-38;
c:=59;


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^2-38x+59

f(0)=59
f(1)=11
f(2)=13
f(3)=23
f(4)=7
f(5)=53
f(6)=19
f(7)=79
f(8)=181
f(9)=101
f(10)=17
f(11)=1
f(12)=1
f(13)=1
f(14)=277
f(15)=1
f(16)=293
f(17)=149
f(18)=43
f(19)=151
f(20)=1
f(21)=1
f(22)=1
f(23)=1
f(24)=1
f(25)=1
f(26)=1
f(27)=1
f(28)=1
f(29)=1
f(30)=1
f(31)=1
f(32)=1
f(33)=1
f(34)=1
f(35)=1
f(36)=1
f(37)=1
f(38)=1
f(39)=1
f(40)=139
f(41)=1
f(42)=227
f(43)=137
f(44)=1
f(45)=1
f(46)=61
f(47)=241
f(48)=1
f(49)=1
f(50)=659
f(51)=1
f(52)=787
f(53)=1
f(54)=71
f(55)=1
f(56)=97
f(57)=571
f(58)=1
f(59)=1
f(60)=197
f(61)=1
f(62)=1
f(63)=1
f(64)=1723
f(65)=907
f(66)=1907
f(67)=1
f(68)=2099
f(69)=157
f(70)=1
f(71)=1201
f(72)=109
f(73)=1307
f(74)=389
f(75)=1
f(76)=421
f(77)=1531
f(78)=1
f(79)=1
f(80)=263
f(81)=1
f(82)=193
f(83)=271
f(84)=3923
f(85)=2027
f(86)=1
f(87)=2161
f(88)=1
f(89)=1
f(90)=677
f(91)=2441
f(92)=457
f(93)=199
f(94)=5323
f(95)=1
f(96)=331
f(97)=1
f(98)=5939
f(99)=3049

b) Substitution of the polynom
The polynom f(x)=x^2-38x+59  could be written as f(y)= y^2-302 with x=y+19

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-19
f'(x)>2x-39  with x > 17

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 : 59, 11, 13, 23, 7, 53, 19, 79, 181, 101, 17, 1, 1, 1, 277, 1, 293, 149, 43, 151,
Found in Database : 59, 11, 13, 23, 7, 53, 19, 79, 181, 101, 17, 277, 293, 149, 43, 151,
Found in Database : 7, 11, 13, 17, 19, 23, 43, 53, 59, 61, 71, 79, 97, 101, 109, 137, 139, 149,