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Development of Algorithmic Constructions

13:52:55 20.Apr 2024

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Polynom = x^2+120x-557

1. Algorithm

a:=1;
b:=120;
c:=-557;


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+120x-557

f(0)=557
f(1)=109
f(2)=313
f(3)=47
f(4)=61
f(5)=17
f(6)=199
f(7)=83
f(8)=467
f(9)=151
f(10)=743
f(11)=13
f(12)=79
f(13)=293
f(14)=1319
f(15)=367
f(16)=1619
f(17)=443
f(18)=41
f(19)=521
f(20)=2243
f(21)=601
f(22)=1
f(23)=683
f(24)=223
f(25)=59
f(26)=1
f(27)=853
f(28)=211
f(29)=941
f(30)=3943
f(31)=1031
f(32)=73
f(33)=1123
f(34)=4679
f(35)=1217
f(36)=5059
f(37)=101
f(38)=419
f(39)=1
f(40)=5843
f(41)=1511
f(42)=6247
f(43)=1613
f(44)=6659
f(45)=1
f(46)=7079
f(47)=1823
f(48)=7507
f(49)=1931
f(50)=1
f(51)=157
f(52)=8387
f(53)=2153
f(54)=8839
f(55)=2267
f(56)=547
f(57)=2383
f(58)=9767
f(59)=1
f(60)=10243
f(61)=2621
f(62)=631
f(63)=1
f(64)=863
f(65)=1
f(66)=11719
f(67)=1
f(68)=12227
f(69)=3121
f(70)=12743
f(71)=3251
f(72)=13267
f(73)=1
f(74)=13799
f(75)=3517
f(76)=1103
f(77)=281
f(78)=14887
f(79)=1
f(80)=15443
f(81)=3931
f(82)=16007
f(83)=4073
f(84)=1
f(85)=4217
f(86)=17159
f(87)=4363
f(88)=17747
f(89)=347
f(90)=1
f(91)=1
f(92)=18947
f(93)=4813
f(94)=19559
f(95)=4967
f(96)=1187
f(97)=1
f(98)=20807
f(99)=5281

b) Substitution of the polynom
The polynom f(x)=x^2+120x-557  could be written as f(y)= y^2-4157 with x=y-60

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+60
f'(x)>2x+119

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 : 557, 109, 313, 47, 61, 17, 199, 83, 467, 151, 743, 13, 79, 293, 1319, 367, 1619, 443, 41, 521,
Found in Database : 557, 109, 313, 47, 61, 17, 199, 83, 467, 151, 743, 13, 79, 293, 1319, 367, 1619, 443, 41, 521, 2243, 601, 683, 223, 59, 853, 211, 941, 3943, 1031, 73, 1123, 4679, 1217, 5059, 101, 419,
Found in Database : 13, 17, 41, 47, 59, 61, 73, 79, 83, 101, 109,