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Development of Algorithmic Constructions

08:51:46 20.Apr 2024

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Polynom = x^2+62x-181

1. Algorithm

a:=1;
b:=62;
c:=-181;


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+62x-181

f(0)=181
f(1)=59
f(2)=53
f(3)=7
f(4)=83
f(5)=11
f(6)=227
f(7)=151
f(8)=379
f(9)=229
f(10)=1
f(11)=311
f(12)=101
f(13)=397
f(14)=883
f(15)=487
f(16)=97
f(17)=1
f(18)=1259
f(19)=1
f(20)=1459
f(21)=71
f(22)=1667
f(23)=887
f(24)=269
f(25)=997
f(26)=43
f(27)=1
f(28)=2339
f(29)=1229
f(30)=2579
f(31)=193
f(32)=257
f(33)=211
f(34)=3083
f(35)=1607
f(36)=3347
f(37)=1741
f(38)=47
f(39)=1879
f(40)=557
f(41)=1
f(42)=79
f(43)=197
f(44)=4483
f(45)=331
f(46)=4787
f(47)=353
f(48)=5099
f(49)=239
f(50)=5419
f(51)=2791
f(52)=821
f(53)=2957
f(54)=1
f(55)=1
f(56)=6427
f(57)=3301
f(58)=6779
f(59)=1
f(60)=1
f(61)=523
f(62)=7507
f(63)=3847
f(64)=7883
f(65)=367
f(66)=1181
f(67)=4231
f(68)=1237
f(69)=103
f(70)=9059
f(71)=421
f(72)=9467
f(73)=691
f(74)=9883
f(75)=1
f(76)=937
f(77)=5261
f(78)=10739
f(79)=5479
f(80)=1597
f(81)=5701
f(82)=1
f(83)=5927
f(84)=281
f(85)=131
f(86)=12547
f(87)=1
f(88)=277
f(89)=947
f(90)=13499
f(91)=6871
f(92)=1
f(93)=647
f(94)=2069
f(95)=139
f(96)=2141
f(97)=7621
f(98)=1409
f(99)=7879

b) Substitution of the polynom
The polynom f(x)=x^2+62x-181  could be written as f(y)= y^2-1142 with x=y-31

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+31
f'(x)>2x+61

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 : 181, 59, 53, 7, 83, 11, 227, 151, 379, 229, 1, 311, 101, 397, 883, 487, 97, 1, 1259, 1,
Found in Database : 181, 59, 53, 7, 83, 11, 227, 151, 379, 229, 311, 101, 397, 883, 487, 97, 1259, 1459, 71, 1667, 887, 269, 997, 43, 2339, 1229, 2579, 193, 257, 211, 3083, 1607, 3347, 1741, 47, 1879,
Found in Database : 7, 11, 43, 47, 53, 59, 71, 79, 83, 97, 101, 103, 131, 139,