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Development of Algorithmic Constructions

10:33:24 19.Apr 2024

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Polynom = x^2+62x-277

1. Algorithm

a:=1;
b:=62;
c:=-277;


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+62x-277

f(0)=277
f(1)=107
f(2)=149
f(3)=41
f(4)=13
f(5)=29
f(6)=131
f(7)=103
f(8)=283
f(9)=181
f(10)=443
f(11)=263
f(12)=47
f(13)=349
f(14)=787
f(15)=439
f(16)=971
f(17)=1
f(18)=1163
f(19)=631
f(20)=1
f(21)=733
f(22)=1571
f(23)=839
f(24)=1787
f(25)=73
f(26)=2011
f(27)=1063
f(28)=2243
f(29)=1181
f(30)=191
f(31)=1303
f(32)=2731
f(33)=1429
f(34)=1
f(35)=1559
f(36)=3251
f(37)=1693
f(38)=271
f(39)=1831
f(40)=3803
f(41)=1973
f(42)=4091
f(43)=163
f(44)=1
f(45)=2269
f(46)=4691
f(47)=2423
f(48)=5003
f(49)=89
f(50)=5323
f(51)=211
f(52)=5651
f(53)=2909
f(54)=5987
f(55)=3079
f(56)=487
f(57)=3253
f(58)=1
f(59)=1
f(60)=7043
f(61)=3613
f(62)=7411
f(63)=1
f(64)=599
f(65)=3989
f(66)=8171
f(67)=1
f(68)=8563
f(69)=337
f(70)=8963
f(71)=4583
f(72)=9371
f(73)=4789
f(74)=9787
f(75)=4999
f(76)=10211
f(77)=401
f(78)=367
f(79)=5431
f(80)=11083
f(81)=5653
f(82)=887
f(83)=5879
f(84)=11987
f(85)=1
f(86)=12451
f(87)=6343
f(88)=12923
f(89)=6581
f(90)=1031
f(91)=6823
f(92)=479
f(93)=7069
f(94)=14387
f(95)=563
f(96)=14891
f(97)=7573
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+62x-277  could be written as f(y)= y^2-1238 with x=y-31

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+31
f'(x)>2x+61

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 : 277, 107, 149, 41, 13, 29, 131, 103, 283, 181, 443, 263, 47, 349, 787, 439, 971, 1, 1163, 631,
Found in Database : 277, 107, 149, 41, 13, 29, 131, 103, 283, 181, 443, 263, 47, 349, 787, 439, 971, 1163, 631, 733, 1571, 839, 1787, 73, 2011, 1063, 2243, 1181, 191, 1303, 2731, 1429, 1559, 3251, 1693, 271, 1831,
Found in Database : 13, 29, 41, 47, 73, 89, 103, 107, 131, 149,