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Development of Algorithmic Constructions

17:46:25 20.Apr 2024

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Polynom = x^2+172x-397

1. Algorithm

a:=1;
b:=172;
c:=-397;


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+172x-397

f(0)=397
f(1)=7
f(2)=1
f(3)=1
f(4)=307
f(5)=61
f(6)=11
f(7)=107
f(8)=149
f(9)=1
f(10)=1423
f(11)=101
f(12)=1811
f(13)=251
f(14)=2207
f(15)=43
f(16)=373
f(17)=1
f(18)=3023
f(19)=1
f(20)=313
f(21)=457
f(22)=79
f(23)=73
f(24)=59
f(25)=283
f(26)=4751
f(27)=311
f(28)=1
f(29)=97
f(30)=809
f(31)=67
f(32)=6131
f(33)=199
f(34)=6607
f(35)=1
f(36)=1013
f(37)=131
f(38)=7583
f(39)=89
f(40)=137
f(41)=521
f(42)=71
f(43)=1
f(44)=1301
f(45)=1171
f(46)=9631
f(47)=1237
f(48)=10163
f(49)=163
f(50)=139
f(51)=1
f(52)=11251
f(53)=1
f(54)=11807
f(55)=1511
f(56)=1
f(57)=113
f(58)=1
f(59)=827
f(60)=13523
f(61)=157
f(62)=103
f(63)=1801
f(64)=191
f(65)=1
f(66)=1
f(67)=1
f(68)=15923
f(69)=2029
f(70)=233
f(71)=1
f(72)=223
f(73)=1093
f(74)=17807
f(75)=1
f(76)=18451
f(77)=2347
f(78)=2729
f(79)=347
f(80)=19763
f(81)=1
f(82)=20431
f(83)=1
f(84)=21107
f(85)=383
f(86)=1
f(87)=2767
f(88)=22483
f(89)=1427
f(90)=239
f(91)=1471
f(92)=3413
f(93)=433
f(94)=2237
f(95)=3121
f(96)=1
f(97)=1
f(98)=389
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+172x-397  could be written as f(y)= y^2-7793 with x=y-86

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+86
f'(x)>2x+171

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 : 397, 7, 1, 1, 307, 61, 11, 107, 149, 1, 1423, 101, 1811, 251, 2207, 43, 373, 1, 3023, 1,
Found in Database : 397, 7, 307, 61, 11, 107, 149, 1423, 101, 1811, 251, 2207, 43, 373, 3023, 313, 457, 79, 73, 59, 283, 4751, 311, 97, 809, 67, 6131, 199, 6607, 1013, 131, 7583, 89,
Found in Database : 7, 11, 43, 59, 61, 67, 71, 73, 79, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149,