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Development of Algorithmic Constructions

00:37:30 20.Apr 2024

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Polynom = x^2+40x-607

1. Algorithm

a:=1;
b:=40;
c:=-607;


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+40x-607

f(0)=607
f(1)=283
f(2)=523
f(3)=239
f(4)=431
f(5)=191
f(6)=331
f(7)=139
f(8)=223
f(9)=83
f(10)=107
f(11)=23
f(12)=17
f(13)=41
f(14)=149
f(15)=109
f(16)=1
f(17)=181
f(18)=19
f(19)=257
f(20)=593
f(21)=337
f(22)=757
f(23)=421
f(24)=929
f(25)=509
f(26)=1109
f(27)=601
f(28)=1297
f(29)=1
f(30)=1493
f(31)=797
f(32)=1697
f(33)=53
f(34)=1
f(35)=1009
f(36)=2129
f(37)=59
f(38)=2357
f(39)=1237
f(40)=2593
f(41)=1
f(42)=2837
f(43)=1481
f(44)=3089
f(45)=1609
f(46)=197
f(47)=1741
f(48)=3617
f(49)=1877
f(50)=229
f(51)=2017
f(52)=4177
f(53)=2161
f(54)=1
f(55)=2309
f(56)=251
f(57)=1
f(58)=5077
f(59)=2617
f(60)=5393
f(61)=2777
f(62)=5717
f(63)=173
f(64)=263
f(65)=3109
f(66)=6389
f(67)=193
f(68)=6737
f(69)=3457
f(70)=1
f(71)=3637
f(72)=7457
f(73)=3821
f(74)=7829
f(75)=211
f(76)=8209
f(77)=4201
f(78)=8597
f(79)=4397
f(80)=1
f(81)=4597
f(82)=9397
f(83)=4801
f(84)=577
f(85)=5009
f(86)=1
f(87)=227
f(88)=10657
f(89)=5437
f(90)=11093
f(91)=5657
f(92)=1
f(93)=5881
f(94)=631
f(95)=1
f(96)=1
f(97)=373
f(98)=12917
f(99)=6577

b) Substitution of the polynom
The polynom f(x)=x^2+40x-607  could be written as f(y)= y^2-1007 with x=y-20

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+20
f'(x)>2x+39

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 : 607, 283, 523, 239, 431, 191, 331, 139, 223, 83, 107, 23, 17, 41, 149, 109, 1, 181, 19, 257,
Found in Database : 607, 283, 523, 239, 431, 191, 331, 139, 223, 83, 107, 23, 17, 41, 149, 109, 181, 19, 257, 593, 337, 757, 421, 929, 509, 1109, 601, 1297, 1493, 797, 1697, 53, 1009, 2129, 59, 2357, 1237,
Found in Database : 17, 19, 23, 41, 53, 59, 83, 107, 109, 139, 149,