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Development of Algorithmic Constructions

06:34:10 19.Apr 2024

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Polynom = x^2-280x+1607

1. Algorithm

a:=1;
b:=-280;
c:=1607;


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^2-280x+1607

f(0)=1607
f(1)=83
f(2)=1051
f(3)=97
f(4)=503
f(5)=29
f(6)=37
f(7)=19
f(8)=569
f(9)=13
f(10)=1093
f(11)=1
f(12)=1609
f(13)=233
f(14)=73
f(15)=1
f(16)=2617
f(17)=179
f(18)=3109
f(19)=419
f(20)=3593
f(21)=479
f(22)=313
f(23)=269
f(24)=349
f(25)=149
f(26)=263
f(27)=653
f(28)=5449
f(29)=709
f(30)=71
f(31)=191
f(32)=6329
f(33)=409
f(34)=1
f(35)=67
f(36)=7177
f(37)=1
f(38)=7589
f(39)=487
f(40)=7993
f(41)=1
f(42)=8389
f(43)=1
f(44)=131
f(45)=59
f(46)=9157
f(47)=1
f(48)=733
f(49)=607
f(50)=761
f(51)=1259
f(52)=277
f(53)=1303
f(54)=10597
f(55)=673
f(56)=10937
f(57)=347
f(58)=1
f(59)=1429
f(60)=11593
f(61)=113
f(62)=11909
f(63)=1
f(64)=643
f(65)=773
f(66)=12517
f(67)=1583
f(68)=12809
f(69)=1619
f(70)=13093
f(71)=827
f(72)=461
f(73)=211
f(74)=1049
f(75)=1721
f(76)=1069
f(77)=1753
f(78)=14149
f(79)=223
f(80)=389
f(81)=907
f(82)=14629
f(83)=1
f(84)=1
f(85)=1871
f(86)=15077
f(87)=1
f(88)=15289
f(89)=1
f(90)=15493
f(91)=1949
f(92)=541
f(93)=1973
f(94)=15877
f(95)=499
f(96)=16057
f(97)=1009
f(98)=16229
f(99)=2039

b) Substitution of the polynom
The polynom f(x)=x^2-280x+1607  could be written as f(y)= y^2-17993 with x=y+140

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-140
f'(x)>2x-281  with x > 134

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 : 1607, 83, 1051, 97, 503, 29, 37, 19, 569, 13, 1093, 1, 1609, 233, 73, 1, 2617, 179, 3109, 419,
Found in Database : 1607, 83, 1051, 97, 503, 29, 37, 19, 569, 13, 1093, 1609, 233, 73, 2617, 179, 3109, 419, 3593, 479, 313, 269, 349, 149, 263, 653, 5449, 709, 71, 191, 6329, 409, 67, 7177, 7589, 487,
Found in Database : 13, 19, 29, 37, 59, 67, 71, 73, 83, 97, 113, 131, 149,