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Development of Algorithmic Constructions

09:24:44 20.Apr 2024

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Polynom = x^2+23x-181

1. Algorithm

a:=1;
b:=23;
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+23x-181

f(0)=181
f(1)=157
f(2)=131
f(3)=103
f(4)=73
f(5)=41
f(6)=7
f(7)=29
f(8)=67
f(9)=107
f(10)=149
f(11)=193
f(12)=239
f(13)=1
f(14)=337
f(15)=389
f(16)=443
f(17)=499
f(18)=557
f(19)=617
f(20)=97
f(21)=743
f(22)=809
f(23)=877
f(24)=947
f(25)=1019
f(26)=1093
f(27)=167
f(28)=43
f(29)=1327
f(30)=1409
f(31)=1493
f(32)=1579
f(33)=1667
f(34)=251
f(35)=1
f(36)=1
f(37)=2039
f(38)=2137
f(39)=2237
f(40)=2339
f(41)=349
f(42)=2549
f(43)=2657
f(44)=2767
f(45)=2879
f(46)=1
f(47)=3109
f(48)=461
f(49)=3347
f(50)=3469
f(51)=3593
f(52)=3719
f(53)=3847
f(54)=1
f(55)=587
f(56)=4243
f(57)=151
f(58)=4517
f(59)=4657
f(60)=4799
f(61)=4943
f(62)=727
f(63)=5237
f(64)=5387
f(65)=191
f(66)=5693
f(67)=5849
f(68)=6007
f(69)=881
f(70)=6329
f(71)=1
f(72)=6659
f(73)=6827
f(74)=6997
f(75)=1
f(76)=1049
f(77)=1
f(78)=179
f(79)=7877
f(80)=8059
f(81)=8243
f(82)=8429
f(83)=1231
f(84)=8807
f(85)=8999
f(86)=317
f(87)=229
f(88)=9587
f(89)=9787
f(90)=1427
f(91)=10193
f(92)=10399
f(93)=10607
f(94)=373
f(95)=269
f(96)=11243
f(97)=1637
f(98)=11677
f(99)=11897

b) Substitution of the polynom
The polynom f(x)=x^2+23x-181  could be written as f(y)= y^2-313.25 with x=y-11.5

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+11.5
f'(x)>2x+22

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, 157, 131, 103, 73, 41, 7, 29, 67, 107, 149, 193, 239, 1, 337, 389, 443, 499, 557, 617,
Found in Database : 181, 157, 131, 103, 73, 41, 7, 29, 67, 107, 149, 193, 239, 337, 389, 443, 499, 557, 617, 97, 743, 809, 877, 947, 1019, 1093, 167, 43, 1327, 1409, 1493, 1579, 1667, 251, 2039, 2137, 2237,
Found in Database : 7, 29, 41, 43, 67, 73, 97, 103, 107, 131, 149,