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Development of Algorithmic Constructions

01:20:10 20.Apr 2024

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Polynom = x^2+59x-127

1. Algorithm

a:=1;
b:=59;
c:=-127;


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+59x-127

f(0)=127
f(1)=67
f(2)=5
f(3)=59
f(4)=1
f(5)=193
f(6)=263
f(7)=1
f(8)=409
f(9)=97
f(10)=563
f(11)=643
f(12)=29
f(13)=809
f(14)=179
f(15)=983
f(16)=37
f(17)=233
f(18)=1259
f(19)=271
f(20)=1453
f(21)=1553
f(22)=331
f(23)=1759
f(24)=373
f(25)=1973
f(26)=2083
f(27)=439
f(28)=2309
f(29)=1
f(30)=2543
f(31)=2663
f(32)=557
f(33)=2909
f(34)=607
f(35)=3163
f(36)=89
f(37)=137
f(38)=3559
f(39)=739
f(40)=3833
f(41)=1
f(42)=823
f(43)=4259
f(44)=881
f(45)=157
f(46)=4703
f(47)=971
f(48)=5009
f(49)=1033
f(50)=5323
f(51)=5483
f(52)=1129
f(53)=1
f(54)=239
f(55)=6143
f(56)=107
f(57)=1297
f(58)=6659
f(59)=1367
f(60)=7013
f(61)=7193
f(62)=1
f(63)=7559
f(64)=1549
f(65)=7933
f(66)=8123
f(67)=1663
f(68)=1
f(69)=1741
f(70)=307
f(71)=9103
f(72)=1861
f(73)=257
f(74)=1
f(75)=9923
f(76)=10133
f(77)=2069
f(78)=10559
f(79)=431
f(80)=10993
f(81)=11213
f(82)=2287
f(83)=131
f(84)=2377
f(85)=12113
f(86)=12343
f(87)=503
f(88)=12809
f(89)=2609
f(90)=359
f(91)=13523
f(92)=2753
f(93)=14009
f(94)=2851
f(95)=14503
f(96)=14753
f(97)=3001
f(98)=15259
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+59x-127  could be written as f(y)= y^2-997.25 with x=y-29.5

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+29.5
f'(x)>2x+58

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 : 127, 67, 5, 59, 1, 193, 263, 1, 409, 97, 563, 643, 29, 809, 179, 983, 37, 233, 1259, 271,
Found in Database : 127, 67, 5, 59, 193, 263, 409, 97, 563, 643, 29, 809, 179, 983, 37, 233, 1259, 271, 1453, 1553, 331, 1759, 373, 1973, 2083, 439, 2309, 2543, 2663, 557, 2909, 607, 3163, 89, 137, 3559, 739,
Found in Database : 5, 29, 37, 59, 67, 89, 97, 107, 127, 131, 137,