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Development of Algorithmic Constructions

10:27:30 20.Apr 2024

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Polynom = x^2+70x-3

1. Algorithm

a:=1;
b:=70;
c:=-3;


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+70x-3

f(0)=3
f(1)=17
f(2)=47
f(3)=1
f(4)=293
f(5)=31
f(6)=151
f(7)=67
f(8)=23
f(9)=59
f(10)=797
f(11)=37
f(12)=109
f(13)=269
f(14)=1
f(15)=53
f(16)=1373
f(17)=41
f(18)=1
f(19)=211
f(20)=599
f(21)=1
f(22)=43
f(23)=89
f(24)=751
f(25)=593
f(26)=277
f(27)=1
f(28)=2741
f(29)=239
f(30)=1
f(31)=1
f(32)=1087
f(33)=283
f(34)=3533
f(35)=1
f(36)=1
f(37)=1
f(38)=1367
f(39)=1
f(40)=4397
f(41)=379
f(42)=1567
f(43)=607
f(44)=557
f(45)=431
f(46)=5333
f(47)=229
f(48)=1
f(49)=1
f(50)=1999
f(51)=257
f(52)=373
f(53)=181
f(54)=97
f(55)=859
f(56)=2351
f(57)=1
f(58)=1
f(59)=317
f(60)=113
f(61)=1997
f(62)=101
f(63)=349
f(64)=8573
f(65)=1
f(66)=997
f(67)=1
f(68)=1
f(69)=1
f(70)=1
f(71)=139
f(72)=3407
f(73)=2609
f(74)=1
f(75)=1
f(76)=11093
f(77)=1
f(78)=3847
f(79)=1471
f(80)=1
f(81)=1019
f(82)=733
f(83)=1
f(84)=479
f(85)=1
f(86)=263
f(87)=569
f(88)=13901
f(89)=131
f(90)=4799
f(91)=1831
f(92)=4967
f(93)=421
f(94)=15413
f(95)=653
f(96)=1
f(97)=4049
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+70x-3  could be written as f(y)= y^2-1228 with x=y-35

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+35
f'(x)>2x+69

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 : 3, 17, 47, 1, 293, 31, 151, 67, 23, 59, 797, 37, 109, 269, 1, 53, 1373, 41, 1, 211,
Found in Database : 3, 17, 47, 293, 31, 151, 67, 23, 59, 797, 37, 109, 269, 53, 1373, 41, 211, 599, 43, 89, 751, 593, 277, 2741, 239, 1087, 283, 3533, 1367,
Found in Database : 3, 17, 23, 31, 37, 41, 43, 47, 53, 59, 67, 89, 97, 101, 109, 113, 131, 139, 149,