arctan arctan arctan arctan 2 2000 π = 3 + 8 = 3.25 ( ) 2 8 650 π = 4 = 3.6049 9 550 π = 3 3 30 π = 3.622 264 π = 3.459 3 + 0 7 = 3.4085 < π < 3 + 7 = 3.4286 380 π = 3 + 77 250 = 3.46 5 3.45926 < π < 3.45927
202 π = 864 275 = 3.488 579 3.45926535 < π < 3.45926537 585 π = 355 3 = 3.459292 60 35 699 72 706 00 79 27 722 42 739 50 794 40 853 440 873 707 527 2002 24 600 2009 8 2 5769 8037 74 2009 2 2 6999 9999 3 768 π 882 π 2 x 2 2 = 0 2
3 ABC 2 OB B OB l l A OA A OB l A 2 A 2 OB l A 3 A 3, A 4, A C B : 3
l A A2 A3 B O 2: OA k : OB = A k A k+ : A k+ B OB2 + A k B 2 : OB = A k B A k+ B : A k+ B OB A k+ B = ( ) 2 OB + + OB A k B A k B p k+ = p k = OB A k B p 2 k + + p k A OB 30 96 96 96 p 5, 96 p 2 5 + A OB 30 p = 3 p 5 4
p = 3 > 265 53 p 2 = + 3 + 3 > 2 + 265 53 = 57 53 (57 ) 2 p 3 > + + 57 349450 + 57 53 53 = > 033 53 36 ( ) 349450 4729 > 8 (033 ) 2 p 4 > + + 033 085585 + 033 36 36 = > 9337 36 62 ( ) 085585 9377 > 9 (9337 ) 2 p 5 > + + 9337 875543 + 9337 62 62 = > 9347 62 306 ( ) 875543 > 9357 π < 96 = 96 p 5 < 29376 9347 < 22 7 p k (29 ) 2 p 3 < + + 29 780 780 = p = 3 < 35 780 p 2 = + 3 + 3 < 2 + 35 780 = 29 780 908232 + 29 780 < 823 240 ( ) 908232 2055 < 4 5
p 4 < p 5 < (823 240 (007 66 ) 2 + + 823 240 = ) 2 + + 007 66 = π > 96 = 3380929 + 823 < 007 240 66 ( ) 3380929 20227 < 08405 + 007 < 2097 66 396 ( ) 08405 6055 < 6 96 p 2 5 + > 3806 46494225 π > 46494225 < 24207 2 3806 > 25344 46494225 8069 > 223 7 0 4 arctan 60 π 35 3222254720 arctan arctan arctan Taylor arctan arctan tan tan 3 arctan 4 arctan tan 6
y y π/2 π/2 0 π/2 x 0 x π/2 4: y = arctan x 3: y = tan x y = tan x y y x = cos 2 x x y π/2 y 0 y x x x y = cos 2 y y = arctan x 0 x π/2 6: y = arctan x 5: y = tan x arctan tan y = arctan x x = tan y tan arctan y = x y = arctan x x 7
y = tan x x cos 2 x x y y = tan x (x, y) x y ( 5) cos 2 x x = tan y (x, y) y x ( 6) cos 2 y x = tan y y = arctan x (x, y) x y cos 2 y cos 2 y = tan 2 y + = x 2 + y = arctan x x i x 2 + x 2 + = ( 2i x i ) x + i arctan x f(x) = arctan x f (n) (x) f(x) n x n nx n f () (x) = ( 2i x i ) x + i f (2) (x) = ( ) 2i (x i) 2 (x + i) 2 f (3) (x) = 2 ( ) 2i (x i) 3 (x + i) 3 f (4) (x) = 2 3 ( ) 2i (x i) 4 (x + i) 4 f (5) (x) = 2 3 4 ( ) 2i (x i) 5 (x + i) 5 f (n) (x) = ( )n+ (n )! 2i ( ) (x i) n (x + i) n 8
5 Taylor f(x) x = 0 f(x) f(x) n g n (x) = a 0 + a x + a 2 x 2 + + a n x n a 0, a, a n g n (x) f(x) n x = 0 x = 0 0 a 0 = f(0) x = 0 k k =, 2,, n k!a k = f (k) (0) f(x) x = 0 g n (x) = f(0) 0! + f () (0)! g 0 (x) = sin 0 0! g (x) = sin 0 0! g 2 (x) = sin 0 0! g 3 (x) = sin 0 0! g 4 (x) = sin 0 0! g 5 (x) = sin 0 0! = 0 + cos 0 x = x! + cos 0! + cos 0! + cos 0! + cos 0! x sin 0 x 2 = x 2! x sin 0 2! x sin 0 2! x sin 0 2! x + f (2) (0) 2! f(x) = sin x x 2 cos 0 x 3 = x x3 3! 6 x 2 cos 0 3! x 2 cos 0 3! x 2 + + f (n) (0) x n n! x 3 + sin 0 x 4 = x x3 4! 6 x 3 + sin 0 4! x 4 + cos 0 x 5 = x x3 5! 6 + x5 20 f(x) g (x), g 3 (x), g 5 (x) f(x) 7 g n (x) n f(x) f(x) = f(0) 0! + f () (0)! x + f (2) (0) x 2 + = 2! n=0 f (n) (0) x n n! 9
g (x) g 5 (x) - π 0 g 3 (x) π y = sin x 7: f(x) = sin x g (x), g 3 (x), g 5 (x) f(x) x = 0 Taylor Taylor f(x) = log(x + ) f () (x) = x +, f (2) (x) =! (x + ) 2, f (3) (x) = f (n) (x) = ( )n+ (n )! (x + ) n, 2! (x + ) 3,, log(x + ) = x x2 2 + x3 3 x4 4 + Taylor x x > < x Taylor x arctan Taylor f(x) = arctan x f(0) = 0 n ( ) f (n) (x) = ( )n+ (n )! 2i (x i) n (x + i) n 0
n f (n) (0) = ( )n+ (n )! 2i ( i) n (i) = 0 n ( ( i) ) n (i) n f (n) (0) = 0 n n = 2m + f (2m+) (0) = (2m)! ( ) 2i ( i) 2m+ (i) 2m+ = (2m)! ( ) 2i ( ) m ( i) = (2m)!( ) m ( ) m i arctan x arctan x = x x3 3 + x5 5 x7 7 + Taylor x Taylor 67 674 arctan = π 4 π 4 = 3 + 5 7 + 9 + ( 4 3 + 5 ) 7 + + 00000 3.4594653 6 arctan Taylor
6 r 0 θ r(cos θ + i sin θ) α = r(cos θ + i sin θ) r α θ α 8 α = 0 r = 0 θ α 0 θ α θ + 2nπ (n ) α α α arg α α r, θ α 2 r 2, θ 2 α α 2 = r r 2 (cos θ + i sin θ )(cos θ 2 + i sin θ 2 ) ) = r r 2 ((cos θ cos θ 2 sin θ sin θ 2 ) + i(sin θ cos θ 2 + cos θ sin θ 2 ) ( ) = r r 2 cos(θ + θ 2 ) + i sin(θ + θ 2 ) α α 2 = α α 2 arg(α α 2 ) = arg α + arg α 2 + 2nπ (n ) 3 α = r (cos θ + i sin θ ) r 0 θ 8: 2
7 arctan 706 4 arctan 5 arctan 239 = π 4 arctan 5 arctan 239 arctan 5 = n=0 ( ) n, arctan (2n + )52n+ 239 = ( ) n (2n + )239 2n+ n=0 00 n n + i arctan n n i arctan n 9 5 + i 239 i 5 + i = ( + i)(3 2i) 239 i = i( i)(3 + 2i) 4 (5 + i) 4 (239 i) = ( + i) 4 (3 2i) 4 ( i)( i)(3 + 2i) 4 = i( + i) 3 ( 2 + 2 )(3 2 + 2 2 ) 4 tan θ = n θ = arctan n i θ 0 n 9: arg(n + i) = arctan /n 3
4 arg(5 + i) + arg(239 i) = arg( i) + 3 arg( + i) = π 2 + 3π 4 + 2nπ = π 4 + 2nπ n 4 arctan 5 arctan 239 = π 4 + 2nπ n x x3 3 arctan x x ( x 0). ( 4 5 ) 3 5 3 239 π 4 + 2nπ 4 5 239 + 3 239 3 ( 4 5 ) ( 3 5 3 239 > 4 5 ) 00 00 = 75 00 4 5 239 + 3 239 < 4 3 5 0 + 00 = 8 00 75 00 π 4 + 2nπ 8 00 n 0 4 arctan 5 arctan 239 = π 4 arctan arctan 4
8 arctan 0 arctan 239 4 arctan 55 = π 4 arctan 2 + arctan 3 = π 4 2 arctan 3 + arctan 7 = π 4 (730 ) (748 ) (776 ) 2 arctan 2 arctan 7 = π 4 2 arctan 8 + 8 arctan 57 5 arctan 239 = π 4 6 arctan 8 + 2 arctan 57 + arctan 239 = π 4 44 arctan 57 + 7 arctan 2 arctan 239 ( ) (863 ) (896 ) 682 + 24 arctan 2943 = π 4 ( ) arctan n n arctan 9 983 2 arctan + 32 arctan 49 57 5 arctan 239 + 2 arctan 0443 = π 4 2002 24 7730 arctan a arctan b + a 2 arctan b 2 + + a n arctan b n arctan n Q = arctan log 0 b k k= Q 997 83 arctan + 32 arctan 239 + 2 arctan 68 arctan 5832 023 0443 2 arctan 48482 00 arctan 682638 = π 4 5
8 9 5 5 5 3 3 5 3 5 5 5 5 5 5 7 7 5 7 5 9 9 5 9 5 5 5 3 3 5 3 5 5 5 5 5 5 7 7 5 7 arctan 5 239 239 239 3 3 239 3 239 5 5 239 5 arctan 239 π 6
4 arctan 5 arctan 239 = π 4 2 arctan + 32 arctan 49 57 5 arctan 239 + 2 arctan 0443 = π 4 arctan arctan 9 p, q p + iq 3i, i, 0, 3 7i 4, i,, i p + qi 4 p + qi, q + pi, p qi, q pi p + qi -q + pi -p - qi q - pi 0: 7
4 90 0 π/4 π/4 ± 3 + 4i 2 3 + 4i = ( 2i) 2, 2 = ( + i)( i) 2i + i arctan 2 4n + 2 4n + a 2 + b 2 (a + bi)(a bi) 4 4n + 3 4n + 3 p + qi a + bi c + di p + qi = (a + bi)(c + di) p qi = (a bi)(c di) p 2 + q 2 = (p + qi)(p qi) = (a + bi)(a bi)(c + di)(c di) = (a 2 + b 2 )(c 2 + d 2 ) p + qi a + bi p 2 + q 2 a 2 + b 2 8
2 4n + (a + bi)(a bi) p + qi p 2 + q 2 p 2 + q 2 p 2 + q 2 2 4n + a 2 + b 2 a + bi a bi p + qi 370 + 560i 370 + 560i = 2 5 7 (53 + 8i) 7 4n + 3 5 4n + 5 = 2 2 + 2 5 = (2 + i)(2 i) 2 ( + i)( i) 370 + 560i = 7( + i)( i)(2 + i)(2 i)(53 + 8i) 53 + 8i 53 2 + 8 2 = 2873 = 3 2 7 3 = 3 2 + 2 2 7 = 4 2 + 2 3 + 2i 3 2i 2 4 + i 4 i 53 + 8i 3 + 2i 3 + 2i 53 + 8i 3 2i = (53 + 8i)(3 2i) 3 2 + 2 2 = = (53 + 8i)(3 + 2i) 3 2 + 2 2 = 43 + 30i 3 53 + 8i = (3 2i)( + 0i) 75 82i 3 = + 0i + 0i 3 + 2i 3 2i 3 + 2i + 0i 3 2i = ( + 0i)(3 + 2i) 3 2 + 2 2 = 3 + 52i 3 = + 4i 370 + 560i = 7( + i)( i)(2 + i)(2 i)(3 2i) 2 ( + 4i) 9
7 4n + 3 + i, i, 2 + i, 2 i, 3 2i, + 4i 2 2 2 4n + 53 + 8i 4 + i 4 i + 4i + 4i 4 i + 4i = i(4 i) π/4 π/4 370 + 560i i = i( + i) + 4i = i(4 i) 370 + 560i = 7( + i) 2 (2 + i)(2 i)(3 2i) 2 (4 i) 0 arctan n ± i ±kπ/4 arctan k = 0 2 ± i 0000 ± i 0 2 ± i, 3 ± 2i, 5 ± 4i 2 + i = 2 + i 2 i = 2 i 3 + i = ( + i)(2 i) 3 i = ( i)(2 + i) 5 + i = ( + i)(3 2i) 5 i = ( i)(3 + 2i) 20
7 + i = i( + i)(2 + i) 2 7 i = i( i)(2 i) 2 8 + i = (2 i)(3 + 2i) 8 i = (2 + i)(3 2i) 9 + i = ( + i)(5 4i) 9 i = ( i)(5 + 4i) 8 + i = i(2 i) 2 (3 2i) 8 i = i(2 + i) 2 (3 + 2i) 32 + i = i(2 + i) 2 (5 + 4i) 32 i = i(2 i) 2 (5 4i) 57 + i = i( + i)(2 + i) 3 (3 2i) 57 i = i( i)(2 i) 3 (3 + 2i) 73 + i = i( + i)(2 i)(3 + 2i)(5 + 4i) 73 i = i( i)(2 + i)(3 2i)(5 4i) 239 + i = i( + i)(3 2i) 4 239 i = i( i)(3 + 2i) 4 2943 + i = i( + i)(2 i) 4 (3 2i) 2 (5 + 4i) 2943 i = i( i)(2 + i) 4 (3 + 2i) 2 (5 4i) ±kπ/4 8 ± i, 57 ± i, 73 ± i, 2943 ± i (8 ± i) a (57 ± i) a 2 (73 ± i) a 3 (2943 ± i) a 4 a,, a 4 ± a k a k 2 ± i, 3 ± 2i, 5 ± 4i 2a + 3a 2 a 3 4a 4 = 0 a a 2 + a 3 2a 4 = 0 a 3 + a 4 = 0 2
a 2 a 2 a 3 = a 4 3 5 5 5 a 4 a,, a 4 a = 2, a 2 = 3, a 3 = 5, a 4 = 5 (8 + i) 2 (57 + i) 3 (73 + i) 5 (2943 i) 5 = i 2 (2 i) 24 (3 2i) 2 ( i) 3 ( + i) 3 (2 + i) 9 (3 2i) 3 ( i) 5 ( + i) 5 (2 i) 5 (3 + 2i) 5 (5 + 4i) 5 ( i) 5 ( i) 5 (2 + i) 20 (3 + 2i) 0 (5 4i) 5 = i 25 ( + i) 3 ( 2 + 2 ) 5 (2 2 + 2 ) 29 (3 2 + 2 2 ) 5 (5 2 + 4 2 ) 5 = i( + i) 3 ( 2 + 2 ) 5 (2 2 + 2 ) 29 (3 2 + 2 2 ) 5 (5 2 + 4 2 ) 5 n 2 arg(8 + i) + 3 arg(57 + i) + 5 arg(73 + i) + 5 arg(2943 i) = arg( i) + 3 arg( + i) = π 2 + 3π 4 + 2nπ = π 4 + 2nπ 2 arctan 8 + 3 arctan 57 + 5 arctan 73 5 arctan 2943 = π 4 + 2nπ n x x3 3 arctan x x ( x 0). 2 arctan 8 + 3 arctan 57 + 5 arctan 73 5 arctan 2943 2 8 2 3 8 + 3 3 57 3 3 57 + 5 3 73 5 3 73 5 3 2943 2 2 00 + 3 00 3 00 + 5 00 5 00 5 00 = 33 00 22
n = 0 2 arctan 8 + 3 arctan 57 + 5 arctan 73 5 arctan 2943 2 8 + 3 57 + 5 73 5 2943 + 5 3 2943 3 + 3 50 + 5 50 0 + 5 00 2 00 33 00 π 2 + 2nπ 4 00 2 arctan 8 + 3 arctan 57 + 5 arctan 73 5 arctan 2943 = π 4 57±i, 73±i, 239±i, 2943±i (57 ± i) a (73 ± i) a 2 (239 ± i) a 3 (2943 ± i) a 4 2 ± i, 3 ± 2i, 5 ± 4i 3a a 2 4a 4 = 0 a + a 2 4a 3 2a 4 = 0 a 2 + a 4 = 0 a a 2 a 3 = a 4 a =, a 2 =, a 3 =, a 4 = a 4 (57 + i)(73 i)(239 i)(2943 + i) = ( i)( + i)(2 + i) 3 (3 2i) i( i)(2 + i)(3 2i)(5 4i) ( i)( i)(3 + 2i) 4 i( + i)(2 i) 4 (3 2i) 2 (5 + 4i) 23
= i 4 ( 2 + 2 ) 2 (2 2 + 2 ) 4 (3 2 + 2 2 ) 4 (5 2 + 4 2 ) = ( 2 + 2 ) 2 (2 2 + 2 ) 4 (3 2 + 2 2 ) 4 (5 2 + 4 2 ) n arctan 57 arctan 73 + arctan 239 arctan 2943 = 2nπ n n = 0 π 2 ± i, 5 ± 2i, 9 ± 4i 2 + i = 2 + i 2 i = 2 i 3 + i = ( + i)(2 i) 3 i = ( i)(2 + i) 2 + i = (2 + i)(5 2i) 2 i = (2 i)(5 + 2i) 7 + i = i( + i)(2 + i)(5 + 2i) 7 i = i( i)(2 i)(5 2i) 22 + i = (2 + i)(9 4i) 22 i = (2 i)(9 + 4i) 4 + i = ( + i)(5 2i) 2 4 i = ( i)(5 + 2i) 2 75 + i = i( + i)(5 + 2i)(9 + 4i) 75 i = i( i)(5 2i)(9 4i) 493 + i = i( + i)(2 i) 5 (5 + 2i)(9 4i) 493 i = i( i)(2 + i) 5 (5 2i)(9 + 4i) ±kπ/4 24
(22 + i) 0 (4 + i) 7 (75 + i) 2 (493 + i) 2 = (2 + i) 0 (9 4i) 0 ( + i) 7 (5 2i) 4 ( i) 2 ( + i) 2 (5 + 2i) 2 (9 + 4i) 2 i 2 ( + i) 2 (2 i) 0 (5 + 2i) 2 (9 4i) 2 = i 4 ( + i) 2 (2 2 + 2 ) 0 (5 2 + 2 2 ) 4 (9 2 + 4 2 ) 2 = ( + i) 2 (2 2 + 2 ) 0 (5 2 + 2 2 ) 4 (9 2 + 4 2 ) 2 n 0 arg(22 + i) + 7 arg(4 + i) + 2 arg(75 + i) + 2 arg(493 + i) = arg( ) + 2 arg( + i) = π + 2π 4 + 2nπ = 7π 4 + 2nπ 0 arctan 22 + 7 arctan 4 + 2 arctan 75 + 2 arctan 493 = 7π 4 + 2nπ n x x3 3 arctan x x ( x 0). 0 arctan 22 + 7 arctan + 2 arctan 4 75 + 2 arctan 493 0 22 0 3 22 + 7 3 4 7 3 4 + 2 3 75 2 3 75 + 2 3 493 2 3 493 3 0 50 0 00 + 7 50 7 00 + 2 00 2 00 + 0 00 = 6 00 0 arctan 22 + 7 arctan 4 + 2 arctan 75 + 2 arctan 493 0 22 + 7 4 + 2 75 + 2 493 25
n = 2 0 20 + 7 25 + 2 50 + 00 03 00 6 00 7π 03 + 2nπ 4 00 0 arctan 22 + 7 arctan + 2 arctan 4 75 + 2 arctan 493 = π 4 arctan = π 4 2 arctan 3 + 4 arctan 57 arctan 239 = 5π 4 = kπ 4 + 2nπ n n + i = + i i = i 2 + i = 2 + i 2 i = 2 i 3 + i = ( + i)(2 i) 3 i = ( i)(2 + i) 4 + i = 4 + i 4 i = 4 i 5 + i = ( + i)(3 2i) 26
5 i = ( i)(3 + 2i) 6 + i = 6 + i 6 i = 6 i 7 + i = i( + i)(2 + i) 2 7 i = i( i)(2 i) 2 8 + i = (2 i)(3 + 2i) 8 i = (2 + i)(3 2i) 9 + i = ( + i)(5 4i) 9 i = ( i)(5 + 4i) + i = ( + i)(6 5i) i = ( i)(6 + 5i) 2 + i = (2 + i)(5 2i) 2 i = (2 i)(5 + 2i) 3 + i = ( + i)(2 i)(4 i) 3 i = ( i)(2 + i)(4 + i) 7 + i = i( + i)(2 + i)(5 + 2i) 7 i = i( i)(2 i)(5 2i) 8 + i = i(2 i) 2 (3 2i) 8 i = i(2 + i) 2 (3 + 2i) 2 + i = i( + i)(3 + 2i)(4 + i) 2 i = i( i)(3 2i)(4 i) 22 + i = (2 + i)(9 4i) 22 i = (2 i)(9 + 4i) 23 + i = ( + i)(2 i)(7 2i) 23 i = ( i)(2 + i)(7 + 2i) 27 + i = i( + i)(2 + i)(8 + 3i) 27 i = i( i)(2 i)(8 3i) 30 + i = (4 i)(7 + 2i) 30 i = (4 + i)(7 2i) 3 + i = ( + i)(3 2i)(6 i) 3 i = ( i)(3 + 2i)(6 + i) 32 + i = i(2 + i) 2 (5 + 4i) 32 i = i(2 i) 2 (5 4i) 27
34 + i = (3 + 2i)(8 5i) 34 i = (3 2i)(8 + 5i) 38 + i = (2 i)(4 + i) 2 38 i = (2 + i)(4 i) 2 4 + i = ( + i)(5 2i) 2 4 i = ( i)(5 + 2i) 2 43 + i = ( + i)(2 i) 2 (6 + i) 43 i = ( i)(2 + i) 2 (6 i) 46 + i = (5 + 2i)(8 3i) 46 i = (5 2i)(8 + 3i) 47 + i = i( + i)(2 + i)(3 + 2i)(4 i) 47 i = i( i)(2 i)(3 2i)(4 + i) 50 + i = (5 4i)(6 + 5i) 50 i = (5 + 4i)(6 5i) 55 + i = i( + i)(4 + i)(8 + 5i) 55 i = i( i)(4 i)(8 5i) 57 + i = i( + i)(2 + i) 3 (3 2i) 57 i = i( i)(2 i) 3 (3 + 2i) 68 + i = i(2 i) 3 (6 i) 68 i = i(2 + i) 3 (6 + i) 70 + i = i(3 2i) 2 (5 2i) 70 i = i(3 + 2i) 2 (5 + 2i) 72 + i = (2 + i)(4 + i)(6 5i) 72 i = (2 i)(4 i)(6 + 5i) 73 + i = i( + i)(2 i)(3 + 2i)(5 + 4i) 73 i = i( i)(2 + i)(3 2i)(5 4i) 75 + i = i( + i)(5 + 2i)(9 + 4i) 75 i = i( i)(5 2i)(9 4i) 83 + i = ( + i)(2 i)(3 2i)(7 + 2i) 83 i = ( i)(2 + i)(3 + 2i)(7 2i) 99 + i = i( + i)(3 + 2i) 2 (5 2i) 99 i = i( i)(3 2i) 2 (5 + 2i) 7 + i = i( + i)(2 + i)(6 + i) 2 28
7 i = i( i)(2 i)(6 i) 2 9 + i = ( + i)(8 3i)(9 4i) 9 i = ( i)(8 + 3i)(9 + 4i) 23 + i = ( + i)(2 i)(4 + i)(8 5i) 23 i = ( i)(2 + i)(4 i)(8 + 5i) 32 + i = (2 + i) 2 (4 i)(5 4i) 32 i = (2 i) 2 (4 + i)(5 + 4i) 33 + i = ( + i)(2 i)(5 + 2i)(6 5i) 33 i = ( i)(2 + i)(5 2i)(6 + 5i) 57 + i = i( + i)(2 + i) 2 (4 + i)(5 2i) 57 i = i( i)(2 i) 2 (4 i)(5 + 2i) 72 + i = i(2 + i)(6 + 5i)(9 + 4i) 72 i = i(2 i)(6 5i)(9 4i) 73 + i = ( + i)(2 i)(5 4i)(8 + 3i) 73 i = ( i)(2 + i)(5 + 4i)(8 3i) 82 + i = i(2 + i) 4 (7 2i) 82 i = i(2 i) 4 (7 + 2i) 9 + i = i( + i)(4 + i)(5 + 2i)(6 + i) 9 i = i( i)(4 i)(5 2i)(6 i) 26 + i = (3 + 2i)(6 i)(9 4i) 26 i = (3 2i)(6 + i)(9 + 4i) 233 + i = i( + i)(2 i)(6 + 5i)(8 + 5i) 233 i = i( i)(2 + i)(6 5i)(8 5i) 239 + i = i( + i)(3 2i) 4 239 i = i( i)(3 + 2i) 4 242 + i = i(2 + i)(3 + 2i)(4 + i)(7 + 2i) 242 i = i(2 i)(3 2i)(4 i)(7 2i) 255 + i = ( + i)(3 + 2i)(5 4i)(6 5i) 255 i = ( i)(3 2i)(5 + 4i)(6 + 5i) 265 + i = ( + i)(3 2i)(6 + i)(8 3i) 265 i = ( i)(3 + 2i)(6 i)(8 + 3i) 268 + i = (2 i) 2 (3 + 2i) 2 (4 i) 268 i = (2 + i) 2 (3 2i) 2 (4 + i) 29
278 + i = (2 i)(3 2i)(5 + 2i)(5 + 4i) 278 i = (2 + i)(3 + 2i)(5 2i)(5 4i) 302 + i = (2 + i)(4 i)(5 2i)(6 + i) 302 i = (2 i)(4 + i)(5 + 2i)(6 i) 307 + i = ( + i)(2 + i) 3 (3 + 2i)(5 + 2i) 307 i = ( i)(2 i) 3 (3 2i)(5 2i) 39 + i = i( + i)(4 i)(5 + 4i)(8 + 3i) 39 i = i( i)(4 + i)(5 4i)(8 3i) 327 + i = i( + i)(2 + i)(4 + i) 2 (6 i) 327 i = i( i)(2 i)(4 i) 2 (6 + i) 378 + i = i(2 i)(4 + i)(5 4i) 2 378 i = i(2 + i)(4 i)(5 + 4i) 2 40 + i = i( + i)(6 i)(5 + 4i)(7 + 2i) 40 i = i( i)(6 + i)(5 4i)(7 2i) 4 + i = i( + i)(3 + 2i)(8 3i)(8 + 5i) 4 i = i( i)(3 2i)(8 + 3i)(8 5i) 438 + i = i(2 i)(4 i)(6 i)(6 5i) 438 i = i(2 + i)(4 + i)(6 + i)(6 + 5i) 447 + i = ( + i)(2 + i)(3 2i)(5 2i)(7 2i) 447 i = ( i)(2 i)(3 + 2i)(5 + 2i)(7 + 2i) 463 + i = i( + i)(2 i)(3 + 2i)(4 + i)(9 + 4i) 463 i = i( i)(2 + i)(3 2i)(4 i)(9 4i) 500 + i = (7 2i) 2 (8 + 5i) 500 i = (7 + 2i) 2 (8 5i) 507 + i = i( + i)(2 + i) 2 (7 + 2i)(9 4i) 507 i = i( i)(2 i) 2 (7 2i)(9 + 4i) 538 + i = (2 i)(3 2i)(6 + 5i)(8 + 3i) 538 i = (2 + i)(3 + 2i)(6 5i)(8 3i) 557 + i = i( + i)(2 + i) 3 (4 i)(8 3i) 557 i = i( i)(2 i) 3 (4 + i)(8 + 3i) 560 + i = (7 + 2i)(6 5i)(9 + 4i) 560 i = (7 2i)(6 + 5i)(9 4i) 568 + i = i(2 i) 3 (5 + 2i)(8 5i) 30
568 i = i(2 + i) 3 (5 2i)(8 + 5i) 606 + i = i(3 + 2i) 2 (5 + 4i)(7 2i) 606 i = i(3 2i) 2 (5 4i)(7 + 2i) 657 + i = i( + i)(2 + i) 2 (8 5i)(9 + 4i) 657 i = i( i)(2 i) 2 (8 + 5i)(9 4i) 682 + i = (2 + i) 3 (6 5i) 2 682 i = (2 i) 3 (6 + 5i) 2 684 + i = i(3 + 2i)(4 + i)(5 + 2i)(8 + 3i) 684 i = i(3 2i)(4 i)(5 2i)(8 3i) 746 + i = i(3 2i) 2 (6 + i)(8 5i) 746 i = i(3 + 2i) 2 (6 i)(8 + 5i) 829 + i = ( + i)(4 i) 2 (5 + 2i)(5 4i) 829 i = ( i)(4 + i) 2 (5 2i)(5 + 4i) 882 + i = (2 + i) 2 (5 2i) 2 (6 i) 882 i = (2 i) 2 (5 + 2i) 2 (6 + i) 93 + i = i( + i)(3 + 2i)(4 i)(6 + i)(7 + 2i) 93 i = i( i)(3 2i)(4 + i)(6 i)(7 2i) 993 + i = i( + i)(2 i) 2 (3 2i)(6 i)(5 4i) 993 i = i( i)(2 + i) 2 (3 + 2i)(6 + i)(5 + 4i) 068 + i = (2 i) 6 (8 3i) 068 i = (2 + i) 6 (8 + 3i) 43 + i = ( + i)(2 i) 2 (4 + i)(5 2i)(7 + 2i) 43 i = ( i)(2 + i) 2 (4 i)(5 + 2i)(7 2i) 433 + i = ( + i)(2 i)(5 2i)(8 3i)(9 + 4i) 433 i = ( i)(2 + i)(5 + 2i)(8 + 3i)(9 4i) 560 + i = (4 i)(6 + i)(7 2i)(8 + 3i) 560 i = (4 + i)(6 i)(7 + 2i)(8 3i) 568 + i = (2 i) 3 (3 + 2i)(4 + i)(8 + 5i) 568 i = (2 + i) 3 (3 2i)(4 i)(8 5i) 636 + i = (4 + i)(5 2i)(6 + 5i)(8 5i) 636 i = (4 i)(5 + 2i)(6 5i)(8 + 5i) 772 + i = (2 + i)(4 + i) 2 (5 4i)(7 2i) 772 i = (2 i)(4 i) 2 (5 + 4i)(7 + 2i) 3
98 + i = (2 i) 2 (6 i)(5 + 4i)(9 + 4i) 98 i = (2 + i) 2 (6 + i)(5 4i)(9 4i) 2059 + i = i( + i)(3 2i)(5 4i) 2 (9 4i) 2059 i = i( i)(3 + 2i)(5 + 4i) 2 (9 + 4i) 263 + i = ( + i)(2 i)(3 2i)(4 + i)(5 + 2i)(8 3i) 263 i = ( i)(2 + i)(3 + 2i)(4 i)(5 2i)(8 + 3i) 2309 + i = i( + i)(3 + 2i)(7 + 2i) 2 (8 3i) 2309 i = i( i)(3 2i)(7 2i) 2 (8 + 3i) 2436 + i = i(3 2i) 3 (6 i)(8 + 3i) 2436 i = i(3 + 2i) 3 (6 + i)(8 3i) 2673 + i = i( + i)(2 i)(3 + 2i)(4 + i)(7 2i)(6 + 5i) 2673 i = i( i)(2 + i)(3 2i)(4 i)(7 + 2i)(6 5i) 2738 + i = (2 i)(3 + 2i)(5 2i)(5 + 4i)(9 4i) 2738 i = (2 + i)(3 2i)(5 + 2i)(5 4i)(9 + 4i) 297 + i = i( + i)(2 + i)(3 2i)(5 + 2i)(6 i)(6 + 5i) 297 i = i( i)(2 i)(3 + 2i)(5 2i)(6 + i)(6 5i) 2943 + i = i( + i)(2 i) 4 (3 2i) 2 (5 + 4i) 2943 i = i( i)(2 + i) 4 (3 + 2i) 2 (5 4i) 3039 + i = i( + i)(4 i)(6 + 5i) 2 (8 3i) 3039 i = i( i)(4 + i)(6 5i) 2 (8 + 3i) 3793 + i = ( + i)(2 i) 2 (7 + 2i)(6 5i)(8 + 5i) 3793 i = ( i)(2 + i) 2 (7 2i)(6 + 5i)(8 5i) 493 + i = i( + i)(2 i) 5 (5 + 2i)(9 4i) 493 i = i( i)(2 + i) 5 (5 2i)(9 + 4i) 427 + i = ( + i)(2 + i)(3 2i)(5 2i)(7 + 2i)(8 5i) 427 i = ( i)(2 i)(3 + 2i)(5 + 2i)(7 2i)(8 + 5i) 4246 + i = (3 + 2i)(4 i)(5 2i) 2 (9 + 4i) 4246 i = (3 2i)(4 + i)(5 + 2i) 2 (9 4i) 4594 + i = (3 2i)(4 + i)(5 2i)(6 + i)(8 + 5i) 4594 i = (3 + 2i)(4 i)(5 + 2i)(6 i)(8 5i) 4662 + i = i(2 + i)(3 + 2i) 2 (4 + i) 2 (8 5i) 4662 i = i(2 i)(3 2i) 2 (4 i) 2 (8 + 5i) 4747 + i = ( + i)(2 + i)(4 + i)(5 + 4i)(7 + 2i)(6 + 5i) 32
4747 i = ( i)(2 i)(4 i)(5 4i)(7 2i)(6 5i) 4952 + i = (2 + i)(6 i)(5 + 4i)(7 2i)(6 5i) 4952 i = (2 i)(6 + i)(5 4i)(7 + 2i)(6 + 5i) 5257 + i = ( + i)(2 + i) 2 (3 2i)(4 + i)(5 4i)(6 5i) 5257 i = ( i)(2 i) 2 (3 + 2i)(4 i)(5 + 4i)(6 + 5i) 5357 + i = i( + i)(2 + i) 2 (6 + 5i)(9 4i) 2 5357 i = i( i)(2 i) 2 (6 5i)(9 + 4i) 2 5507 + i = ( + i)(2 + i) 2 (3 + 2i) 2 (6 i)(9 + 4i) 5507 i = ( i)(2 i) 2 (3 2i) 2 (6 + i)(9 4i) 5648 + i = (2 i)(4 + i)(7 + 2i)(8 + 3i)(9 4i) 5648 i = (2 + i)(4 i)(7 2i)(8 3i)(9 + 4i) 5667 + i = ( + i)(2 + i)(5 2i)(6 + i)(5 4i)(8 3i) 5667 i = ( i)(2 i)(5 + 2i)(6 i)(5 + 4i)(8 + 3i) 607 + i = ( + i)(2 + i) 2 (4 + i) 2 (5 + 2i)(8 + 5i) 607 i = ( i)(2 i) 2 (4 i) 2 (5 2i)(8 5i) 68 + i = i(2 i) 2 (3 + 2i)(5 4i)(7 2i) 2 68 i = i(2 + i) 2 (3 2i)(5 + 4i)(7 + 2i) 2 6962 + i = i(2 + i)(6 + i) 2 (8 + 3i)(9 + 4i) 6962 i = i(2 i)(6 i) 2 (8 3i)(9 4i) 8368 + i = i(2 i) 2 (4 + i)(6 + i)(6 5i)(8 3i) 8368 i = i(2 + i) 2 (4 i)(6 i)(6 + 5i)(8 + 3i) 993 + i = i( + i)(2 i) 4 (4 i)(5 4i)(9 + 4i) 993 i = i( i)(2 + i) 4 (4 + i)(5 + 4i)(9 4i) 9466 + i = i(5 2i)(6 i) 3 (6 5i) 9466 i = i(5 + 2i)(6 + i) 3 (6 + 5i) 9872 + i = (2 + i)(3 2i) 2 (5 2i)(5 + 4i)(9 + 4i) 9872 i = (2 i)(3 + 2i) 2 (5 + 2i)(5 4i)(9 4i) 33