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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 you are interested in some better algorithms have a look at quadr_Sieb_x^2+1.php.
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^2-208x+863
f(0)=863
f(1)=41
f(2)=11
f(3)=31
f(4)=47
f(5)=19
f(6)=349
f(7)=17
f(8)=67
f(9)=29
f(10)=1117
f(11)=163
f(12)=1489
f(13)=1
f(14)=109
f(15)=127
f(16)=1
f(17)=149
f(18)=2557
f(19)=1
f(20)=2897
f(21)=383
f(22)=3229
f(23)=53
f(24)=1
f(25)=1
f(26)=73
f(27)=503
f(28)=4177
f(29)=541
f(30)=37
f(31)=1
f(32)=251
f(33)=307
f(34)=1
f(35)=59
f(36)=1
f(37)=683
f(38)=193
f(39)=179
f(40)=5857
f(41)=1
f(42)=1
f(43)=1
f(44)=6353
f(45)=809
f(46)=599
f(47)=419
f(48)=401
f(49)=433
f(50)=227
f(51)=1
f(52)=659
f(53)=919
f(54)=257
f(55)=1
f(56)=7649
f(57)=1
f(58)=461
f(59)=991
f(60)=8017
f(61)=1013
f(62)=431
f(63)=1
f(64)=8353
f(65)=1
f(66)=1
f(67)=1
f(68)=787
f(69)=1091
f(70)=463
f(71)=277
f(72)=8929
f(73)=281
f(74)=823
f(75)=1
f(76)=173
f(77)=1153
f(78)=9277
f(79)=1
f(80)=9377
f(81)=1
f(82)=557
f(83)=1
f(84)=233
f(85)=1
f(86)=9629
f(87)=151
f(88)=9697
f(89)=1
f(90)=887
f(91)=1223
f(92)=577
f(93)=1229
f(94)=167
f(95)=617
f(96)=1
f(97)=619
f(98)=211
f(99)=1
b) Substitution of the polynom
The polynom f(x)=x^2-208x+863 could be written as f(y)= y^2-9953 with x=y+104
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-104
f'(x)>2x-209 with x > 100