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Development of Algorithmic Constructions

23:41:08 19.Apr 2024

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Polynom = x^2+28x-277

1. Algorithm

a:=1;
b:=28;
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+28x-277

f(0)=277
f(1)=31
f(2)=7
f(3)=23
f(4)=149
f(5)=1
f(6)=73
f(7)=1
f(8)=11
f(9)=1
f(10)=103
f(11)=19
f(12)=29
f(13)=1
f(14)=311
f(15)=1
f(16)=61
f(17)=1
f(18)=1
f(19)=1
f(20)=683
f(21)=47
f(22)=823
f(23)=1
f(24)=971
f(25)=131
f(26)=1
f(27)=151
f(28)=1291
f(29)=43
f(30)=1
f(31)=97
f(32)=53
f(33)=1
f(34)=1831
f(35)=241
f(36)=2027
f(37)=1
f(38)=1
f(39)=1
f(40)=349
f(41)=1
f(42)=2663
f(43)=347
f(44)=59
f(45)=1
f(46)=1
f(47)=1
f(48)=3371
f(49)=1
f(50)=3623
f(51)=67
f(52)=353
f(53)=251
f(54)=593
f(55)=1
f(56)=233
f(57)=571
f(58)=673
f(59)=607
f(60)=5003
f(61)=1
f(62)=5303
f(63)=1
f(64)=181
f(65)=1
f(66)=5927
f(67)=761
f(68)=1
f(69)=401
f(70)=227
f(71)=211
f(72)=1
f(73)=887
f(74)=661
f(75)=1
f(76)=263
f(77)=1
f(78)=1
f(79)=1
f(80)=8363
f(81)=1069
f(82)=1249
f(83)=1117
f(84)=397
f(85)=1
f(86)=1361
f(87)=1
f(88)=9931
f(89)=1
f(90)=10343
f(91)=1319
f(92)=229
f(93)=1
f(94)=1
f(95)=1
f(96)=1
f(97)=1481
f(98)=12071
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+28x-277  could be written as f(y)= y^2-473 with x=y-14

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+14
f'(x)>2x+27

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, 31, 7, 23, 149, 1, 73, 1, 11, 1, 103, 19, 29, 1, 311, 1, 61, 1, 1, 1,
Found in Database : 277, 31, 7, 23, 149, 73, 11, 103, 19, 29, 311, 61, 683, 47, 823, 971, 131, 151, 1291, 43, 97, 53, 1831, 241, 2027,
Found in Database : 7, 11, 19, 23, 29, 31, 43, 47, 53, 59, 61, 67, 73, 97, 103, 131, 149,