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Development of Algorithmic Constructions

11:03:35 20.Apr 2024

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Polynom = x^2+180x-277

1. Algorithm

a:=1;
b:=180;
c:=-277;


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+180x-277

f(0)=277
f(1)=3
f(2)=29
f(3)=17
f(4)=1
f(5)=1
f(6)=839
f(7)=43
f(8)=409
f(9)=89
f(10)=541
f(11)=19
f(12)=2027
f(13)=31
f(14)=271
f(15)=331
f(16)=953
f(17)=1
f(18)=173
f(19)=73
f(20)=1
f(21)=1
f(22)=463
f(23)=61
f(24)=149
f(25)=101
f(26)=1693
f(27)=83
f(28)=1
f(29)=241
f(30)=317
f(31)=1
f(32)=1
f(33)=211
f(34)=2333
f(35)=151
f(36)=7499
f(37)=1
f(38)=157
f(39)=1033
f(40)=947
f(41)=1
f(42)=109
f(43)=97
f(44)=103
f(45)=1231
f(46)=3373
f(47)=433
f(48)=10667
f(49)=1
f(50)=1
f(51)=719
f(52)=3929
f(53)=503
f(54)=727
f(55)=1
f(56)=227
f(57)=827
f(58)=167
f(59)=1
f(60)=487
f(61)=601
f(62)=4909
f(63)=1879
f(64)=5113
f(65)=163
f(66)=15959
f(67)=113
f(68)=1
f(69)=2113
f(70)=5741
f(71)=1
f(72)=1051
f(73)=379
f(74)=6173
f(75)=1
f(76)=2131
f(77)=1
f(78)=223
f(79)=1
f(80)=6841
f(81)=1
f(82)=7069
f(83)=449
f(84)=359
f(85)=1
f(86)=1
f(87)=1
f(88)=457
f(89)=1
f(90)=24023
f(91)=127
f(92)=1
f(93)=1
f(94)=1
f(95)=1
f(96)=1
f(97)=1
f(98)=1
f(99)=1709

b) Substitution of the polynom
The polynom f(x)=x^2+180x-277  could be written as f(y)= y^2-8377 with x=y-90

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+90
f'(x)>2x+179

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 : 277, 3, 29, 17, 1, 1, 839, 43, 409, 89, 541, 19, 2027, 31, 271, 331, 953, 1, 173, 73,
Found in Database : 277, 3, 29, 17, 839, 43, 409, 89, 541, 19, 2027, 31, 271, 331, 953, 173, 73, 463, 61, 149, 101, 1693, 83, 241, 317, 211, 2333, 151, 7499, 157, 1033,
Found in Database : 3, 17, 19, 29, 31, 43, 61, 73, 83, 89, 97, 101, 103, 109, 113, 127, 139, 149,