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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^2-212x+307
f(0)=307
f(1)=3
f(2)=113
f(3)=5
f(4)=7
f(5)=13
f(6)=929
f(7)=47
f(8)=53
f(9)=19
f(10)=571
f(11)=17
f(12)=23
f(13)=1
f(14)=29
f(15)=331
f(16)=41
f(17)=1
f(18)=1
f(19)=1
f(20)=3533
f(21)=463
f(22)=1291
f(23)=101
f(24)=1
f(25)=1
f(26)=647
f(27)=293
f(28)=1
f(29)=1
f(30)=5153
f(31)=1
f(32)=1
f(33)=1
f(34)=383
f(35)=1
f(36)=6029
f(37)=257
f(38)=97
f(39)=1
f(40)=313
f(41)=419
f(42)=6833
f(43)=1
f(44)=109
f(45)=1
f(46)=349
f(47)=1
f(48)=89
f(49)=1
f(50)=7793
f(51)=1
f(52)=2671
f(53)=1
f(54)=1
f(55)=347
f(56)=8429
f(57)=1
f(58)=1
f(59)=1
f(60)=1259
f(61)=1
f(62)=1
f(63)=227
f(64)=1
f(65)=1
f(66)=491
f(67)=1
f(68)=271
f(69)=239
f(70)=1
f(71)=1213
f(72)=337
f(73)=1
f(74)=283
f(75)=1
f(76)=3343
f(77)=1
f(78)=2029
f(79)=1
f(80)=10253
f(81)=1
f(82)=1
f(83)=1
f(84)=2089
f(85)=1
f(86)=10529
f(87)=1321
f(88)=1
f(89)=1
f(90)=821
f(91)=223
f(92)=10733
f(93)=269
f(94)=719
f(95)=193
f(96)=1
f(97)=1
f(98)=1
f(99)=1
b) Substitution of the polynom
The polynom f(x)=x^2-212x+307 could be written as f(y)= y^2-10929 with x=y+106
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-106
f'(x)>2x-213 with x > 105