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Development of Algorithmic Constructions

13:17:14 20.Apr 2024

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Polynom = x^2+49x-97

1. Algorithm

a:=1;
b:=49;
c:=-97;


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+49x-97

f(0)=97
f(1)=47
f(2)=5
f(3)=59
f(4)=23
f(5)=173
f(6)=233
f(7)=1
f(8)=359
f(9)=17
f(10)=29
f(11)=563
f(12)=127
f(13)=709
f(14)=157
f(15)=863
f(16)=41
f(17)=1
f(18)=1109
f(19)=239
f(20)=1283
f(21)=1373
f(22)=293
f(23)=1559
f(24)=331
f(25)=1753
f(26)=109
f(27)=1
f(28)=71
f(29)=433
f(30)=2273
f(31)=2383
f(32)=499
f(33)=2609
f(34)=1
f(35)=2843
f(36)=2963
f(37)=617
f(38)=3209
f(39)=1
f(40)=3463
f(41)=3593
f(42)=149
f(43)=227
f(44)=1
f(45)=4133
f(46)=4273
f(47)=883
f(48)=1
f(49)=941
f(50)=211
f(51)=5003
f(52)=1031
f(53)=5309
f(54)=1093
f(55)=5623
f(56)=5783
f(57)=1
f(58)=1
f(59)=251
f(60)=379
f(61)=389
f(62)=1
f(63)=6959
f(64)=1427
f(65)=103
f(66)=1
f(67)=307
f(68)=271
f(69)=1609
f(70)=8233
f(71)=8423
f(72)=1723
f(73)=383
f(74)=1801
f(75)=9203
f(76)=9403
f(77)=113
f(78)=577
f(79)=2003
f(80)=10223
f(81)=10433
f(82)=2129
f(83)=10859
f(84)=443
f(85)=491
f(86)=397
f(87)=2347
f(88)=11959
f(89)=2437
f(90)=12413
f(91)=269
f(92)=1
f(93)=13109
f(94)=1
f(95)=1
f(96)=601
f(97)=1
f(98)=349
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+49x-97  could be written as f(y)= y^2-697.25 with x=y-24.5

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+24.5
f'(x)>2x+48

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 : 97, 47, 5, 59, 23, 173, 233, 1, 359, 17, 29, 563, 127, 709, 157, 863, 41, 1, 1109, 239,
Found in Database : 97, 47, 5, 59, 23, 173, 233, 359, 17, 29, 563, 127, 709, 157, 863, 41, 1109, 239, 1283, 1373, 293, 1559, 331, 1753, 109, 71, 433, 2273, 2383, 499, 2609, 2843, 2963, 617, 3209,
Found in Database : 5, 17, 23, 29, 41, 47, 59, 71, 97, 103, 109, 113, 127, 131, 137, 139, 149,