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Development of Algorithmic Constructions

13:22:00 20.Apr 2024

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Polynom = x^2+29x-409

1. Algorithm

a:=1;
b:=29;
c:=-409;


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+29x-409

f(0)=409
f(1)=379
f(2)=347
f(3)=313
f(4)=277
f(5)=239
f(6)=199
f(7)=157
f(8)=113
f(9)=67
f(10)=19
f(11)=31
f(12)=83
f(13)=137
f(14)=193
f(15)=251
f(16)=311
f(17)=373
f(18)=23
f(19)=503
f(20)=571
f(21)=641
f(22)=1
f(23)=787
f(24)=863
f(25)=941
f(26)=1021
f(27)=1103
f(28)=1187
f(29)=1
f(30)=1361
f(31)=1451
f(32)=1543
f(33)=1637
f(34)=1733
f(35)=1831
f(36)=1931
f(37)=107
f(38)=2137
f(39)=2243
f(40)=2351
f(41)=1
f(42)=1
f(43)=2687
f(44)=2803
f(45)=127
f(46)=3041
f(47)=3163
f(48)=173
f(49)=3413
f(50)=3541
f(51)=3671
f(52)=3803
f(53)=1
f(54)=4073
f(55)=4211
f(56)=229
f(57)=4493
f(58)=4637
f(59)=4783
f(60)=4931
f(61)=5081
f(62)=5233
f(63)=5387
f(64)=241
f(65)=5701
f(66)=5861
f(67)=317
f(68)=269
f(69)=6353
f(70)=6521
f(71)=6691
f(72)=6863
f(73)=227
f(74)=7213
f(75)=389
f(76)=1
f(77)=7753
f(78)=7937
f(79)=8123
f(80)=8311
f(81)=8501
f(82)=8693
f(83)=8887
f(84)=293
f(85)=9281
f(86)=499
f(87)=421
f(88)=9887
f(89)=10093
f(90)=10301
f(91)=457
f(92)=10723
f(93)=10937
f(94)=587
f(95)=1
f(96)=1
f(97)=11813
f(98)=12037
f(99)=12263

b) Substitution of the polynom
The polynom f(x)=x^2+29x-409  could be written as f(y)= y^2-619.25 with x=y-14.5

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+14.5
f'(x)>2x+28

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 : 409, 379, 347, 313, 277, 239, 199, 157, 113, 67, 19, 31, 83, 137, 193, 251, 311, 373, 23, 503,
Found in Database : 409, 379, 347, 313, 277, 239, 199, 157, 113, 67, 19, 31, 83, 137, 193, 251, 311, 373, 23, 503, 571, 641, 787, 863, 941, 1021, 1103, 1187, 1361, 1451, 1543, 1637, 1733, 1831, 1931, 107, 2137, 2243,
Found in Database : 19, 23, 31, 67, 83, 107, 113, 127, 137,