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Development of Algorithmic Constructions

01:24:36 20.Apr 2024

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Polynom = x^2+29x-59

1. Algorithm

a:=1;
b:=29;
c:=-59;


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-59

f(0)=59
f(1)=29
f(2)=3
f(3)=37
f(4)=73
f(5)=1
f(6)=151
f(7)=193
f(8)=79
f(9)=283
f(10)=331
f(11)=127
f(12)=433
f(13)=487
f(14)=181
f(15)=601
f(16)=661
f(17)=241
f(18)=787
f(19)=853
f(20)=307
f(21)=991
f(22)=1063
f(23)=379
f(24)=1213
f(25)=1291
f(26)=457
f(27)=1453
f(28)=53
f(29)=541
f(30)=1
f(31)=1801
f(32)=631
f(33)=1987
f(34)=2083
f(35)=727
f(36)=2281
f(37)=2383
f(38)=829
f(39)=2593
f(40)=1
f(41)=937
f(42)=1
f(43)=3037
f(44)=1051
f(45)=3271
f(46)=3391
f(47)=1171
f(48)=3637
f(49)=71
f(50)=1297
f(51)=4021
f(52)=4153
f(53)=1429
f(54)=4423
f(55)=4561
f(56)=1567
f(57)=167
f(58)=4987
f(59)=1
f(60)=5281
f(61)=5431
f(62)=1861
f(63)=5737
f(64)=83
f(65)=2017
f(66)=6211
f(67)=6373
f(68)=2179
f(69)=6703
f(70)=6871
f(71)=2347
f(72)=7213
f(73)=89
f(74)=2521
f(75)=7741
f(76)=1
f(77)=1
f(78)=8287
f(79)=229
f(80)=2887
f(81)=1
f(82)=9043
f(83)=3079
f(84)=9433
f(85)=9631
f(86)=113
f(87)=1
f(88)=353
f(89)=1
f(90)=10651
f(91)=10861
f(92)=3691
f(93)=11287
f(94)=11503
f(95)=3907
f(96)=11941
f(97)=12163
f(98)=4129
f(99)=12613

b) Substitution of the polynom
The polynom f(x)=x^2+29x-59  could be written as f(y)= y^2-269.25 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 : 59, 29, 3, 37, 73, 1, 151, 193, 79, 283, 331, 127, 433, 487, 181, 601, 661, 241, 787, 853,
Found in Database : 59, 29, 3, 37, 73, 151, 193, 79, 283, 331, 127, 433, 487, 181, 601, 661, 241, 787, 853, 307, 991, 1063, 379, 1213, 1291, 457, 1453, 53, 541, 1801, 631, 1987, 2083, 727, 2281, 2383, 829, 2593,
Found in Database : 3, 29, 37, 53, 59, 71, 73, 79, 83, 89, 113, 127, 137,