BBP-type formulas, in general bases, for arctangents of real numbers

Transcription

1 Noes on Number Theory and Discree Mahemaics Vol. 19, 13, No. 3, BBP-ype formulas, in general bases, for arcangens of real numbers Kunle Adegoke 1 and Olawanle Layeni 2 1 Deparmen of Physics, Obafemi Awolowo Universiy Ile-Ife, Nigeria 2 Deparmen of Mahemaics, Obafemi Awolowo Universiy Ile-Ife, Nigeria Absrac: BBP-ype formulas are usually discovered experimenally, one a a ime and in specific bases, hrough compuer searches using PSLQ or oher ineger relaions finding algorihms. In his paper, however, we give a sysemaic analyical derivaion of numerous new explici digi exracion BBP-ype formulas for he arcangens of real numbers in general bases. Our mehod has he clear advanage ha he proofs of he formulas are conained in he derivaions, whereas in he experimenal approach, proofs of discovered formulas have o be found separaely. The high poin of his paper is perhaps he discovery, for he firs ime, of a BBP-ype formula for π 5. Keywords: BBP-ype formulas, Polyalgorihm consans, Digi exracion formulas. AMS Classificaion: 11Y, B99. 1 Inroducion A BBP-ype formula has he remarkable propery ha i allows he i-h digi of a mahemaical consan o be compued wihou having o compue any of he previous i 1 digis and wihou requiring ulra high-precision [1, 2]. BBP-ype formulas were firs inroduced in a 1996 paper [3], where a formula of his ype for π was given. Apar from digi exracion, anoher reason he sudy of BBP-ype formulas has coninued o arac aenion is ha BBP-ype consans are conjecured o be eiher raional or normal o base b [4, 5, 6], ha is heir base-b digis are randomly disribued. BBP-ype formulas are usually discovered experimenally, one a a ime and in specific bases, hrough compuer searches. In his paper, however, we derive explici digi exracion BBP-ype formulas in general bases. Alhough mos of he formulas presened are new, our mehod also accomodaes a rediscovery of known resuls. 33

2 2 Definiions and noaion The firs degree polylogarihm funcion, wih which we are concerned in his paper, is defined by Li 1 [z] = k=1 z k k, z 1. For q, x R, we have he ideniies ( ) q sin x arcan = Im Li 1 [q exp(ix)] = 1 q cos x k 1 q k sin kx k and 1 2 log ( 1 2q cos x + q 2) = Re Li 1 [q exp(ix)] = k 1 The consans considered in his paper are of he form q k cos kx k. K = = n } 1 imπ α m δ gcd(m,n),1 Im Li 1 exp n n ( (1) 1/ sin(mπ/n) α m δ gcd(m,n),1 arcan 1 1/ cos(mπ/n) m=1 m=1 where gcd(j, k) is he greaes common divisor of he inegers j and k, δ jk is Kronecker s dela symbol, n 1, 2, 3, 4, 5, 6, 8, 10, 12,,,,,, }, Z + and α j 1, 0, 1} for each j. Noe ha n here belongs o he se of posiive inegers such ha sin(jπ/n) and cos(jπ/n), j < n Z +, are expressible in erms of radicals. The scheme for obaining he BBP-ype formulas is as follows: For each n, we seek all α j combinaions for which K has a degree 1 BBP-ype formula, ha is, such ha K can be wrien K = 1 l a j b k kl + j, k 0 j=1 where b, l (base, lengh respecively) and a j are inegers. This is accomplished by wriing K = = n } 1 imπ α m δ gcd(m,n),1 Im Li 1 exp n n αm δ ( gcd(m,n),1 rmπ ) } sin r r n m=1 m=1 = 1 r r n m=1 [ ( rmπ )] } α m δ gcd(m,n),1 sin, n and hen choosing only combinaions of α j ha give BBP-ype formulas. 34 (2)

3 In order o save space, we will ofen give he BBP-ype formulas using he compac Bailey s P-noaion [2]: P (s, b, l, A) k 0 1 b k l j=1 a j (kl + j) s, where s, b and l are inegers, and A = (a 1, a 2,..., a l ) is a vecor of inegers. 3 n=1: Base formulas There are no base lengh 1 or lengh 2 arcangen formulas since Li 1 [ 1 / exp iπ] is a real number for > 0. 4 n=2: Base 2 formulas Choosing n = 2 in Eq. (2), we find 1 iπ K = α 1 Im Li 1 exp 2 = 1 [ )] } α 1 sin, r r 2 (3) from which follows immediaely he BBP-ype formula ( ) 1 3 arcan = [ ] 1 ( 2 ) k (4k + 1) 1 (4k + 3) k 0 = P (1, 2, 4, (, 0, 1, 0)) (4) 5 n = 3: Bases 3 and 6 formulas Choosing n = 3 in Eq. (2), we find 1 iπ 1 2iπ K = α 1 Im Li 1 exp + α 2 Im Li 1 exp 3 3 ) ( )]} 2rπ (5) α 1 sin + α 2 sin r r 3 3 All combinaions of α 1, α 2 1, 0, 1} give BBP-ype formulas for K. We herefore consider he combinaions in urn: [1, 0] [0, 1] [α 1, α 2 ] = (6) [1, 1] [1, 1] 35

4 5.1 Case [α 1, α 2 ] = [1, 0] Wih α 1 = 1 = 1 α 2 in Eq. (5), we have 1 iπ K = Im Li 1 exp 3 ( ) 3 = arcan 2 1 = 1 ) r r sin 3 (7) = k 0 k 0 [ 1 1/2 3 ( 3 ) k 6k k + 2 6k + 4 ] 6k + 5 Noe ha Eq. (7) is BBP-ype only if Q. Thus replacing wih 2, we obain he following BBP-ype formula: 2 ( ) 3 3 arcan 2 1 = 3 [ ( 6 ) k 6k k + 2 6k ] (8) 6k + 5 = 3 5 P (1, 6, 6, ( 4, 3, 0,, 1, 0)). Henceforh we shall make such 2 replacemens wihou furher commens, as he need arises. 5.2 Case [α 1, α 2 ] = [0, 1] Wih α 1 = 0 = α 2 1 in Eq. (5), we have 2 ( ) 3 3 arcan = 3 [ 1 5 ( 6 ) k k 0 4 6k k k ] 6k + 5 (9) = 3 5 P (1, 6, 6, ( 4, 3, 0,, 1, 0)). 5.3 Case [α 1, α 2 ] = [1, 1] Wih α 1 = 1 = α 2 in Eq. (5), we have 36

5 ( ) 3 3 arcan 1 = 3 3 k 0 1 ( 3 ) k [ 5 6k + 1 ] 6k + 5 = 3 5 P (1, 3, 6, ( 2, 0, 0, 0, 1, 0)) (10) 5.4 Case [α 1, α 2 ] = [1, 1] Wih α 1 = 1 = α 2 in Eq. (5), we have 2 ( ) 3 3 arcan = 3 [ 1 2 ( 3 ) k 3k ] 3k + 2 k 0 (11) = 3 2 P (1, 3, 3, (, 1, 0)) Noe ha Eq. (11) is he base 3, lengh 3 version of Eq. (9). 6 n=4: Bases 4 and formulas Choosing n = 4 in Eq. (2), we find 1 iπ 1 3iπ K = α 1 Im Li 1 exp + α 3 Im Li 1 exp 4 4 ) ( )]} 3rπ (12) α 1 sin + α 3 sin r r 4 4 As in he previous case of n = 3, all combinaions of α 1, α 3 1, 0, 1} give BBP-ype formulas for K. We will consider, in urn, he combinaions [α 1, α 3 ] = [1, 0], [α 1, α 3 ] = [0, 1], [α 1, α 3 ] = [1, 1] and [α 1, α 3 ] = [1, 1]. 6.1 Case [α 1, α 3 ] = [1, 0] Wih α 1 = 1 = 1 α 3 in Eq. (12), we have ( ) 1 arcan 2 1 = P (1, 168, 8, (8 6, 8 5, 4 4, 0, 2 2, 2, 1, 0)). (13) 37

6 6.2 Case [α 1, α 3 ] = [0, 1] Wih α 1 = 0 = α 3 1 in Eq. (12), we have ( ) 1 arcan = P (1, 168, 8, (8 6, 8 5, 4 4, 0, 2 2, 2, 1, 0)). (14) 6.3 Case [α 1, α 3 ] = [1, 1] Wih α 1 = 1 = α 3 in Eq. (12), we have ( ) 2 2 arcan 1 = 2 7 P (1, 4, 8, ( 3, 0, 2, 0,, 0, 1, 0)). () 6.4 Case [α 1, α 3 ] = [1, 1] Here Eq. (4) is reproduced. 7 n=5: Base 5 formulas Choosing n = 5 in Eq. (2), we find 1 iπ 1 2iπ K = α 1 Im Li 1 exp + α 2 Im Li 1 exp iπ 1 4iπ + α 3 Im Li 1 exp + α 4 Im Li 1 exp 5 5 ) ( ) ( ) ( )]} 2rπ 3rπ 4rπ α 1 sin + α 2 sin + α 3 sin + α 4 sin r r (16) Ou of all he independen combinaions of α j, j = 1, 2, 3, 4, none gives a BBP-ype series for K. 8 n=6: Base 6 formulas Choosing n = 6 in Eq. (2), we find 1 iπ 1 5iπ K = α 1 Im Li 1 exp + α 5 Im Li 1 exp 6 6 ) ( )]} 5rπ (17) α 1 sin + α 5 sin r r 6 6 Ou of he four independen combinaions of α 1, α 5 1, 0, 1}, wo, namely [α 1, α 5 ] = [1, 1] and [α 1, α 5 ] = [1, 1], give BBP-ype formulas for K. 38

7 8.1 Case [α 1, α 5 ] = [1, 1] Wih α 1 = 1 = α 5 in Eq. (17), we have ( ) arcan 1 = 1 11 P (1, 6, 12, ( 5, 0, 2 4, 0, 3, 0, 2, 0, 2, 0, 1, 0)). (18) 8.2 Case [α 1, α 5 ] = [1, 1] Wih α 1 = 1 = α 5 in Eq. (17), Eq. (8) is merely reproduced. 9 n = 8: Base 8 formulas Choosing n = 8 in Eq. (2), we find 1 iπ 1 3iπ K = α 1 Im Li 1 exp + α 3 Im Li 1 exp iπ 1 7iπ + α 5 Im Li 1 exp + α 7 Im Li 1 exp 8 8 ) ( ) ( ) ( )]} 3rπ 5rπ 7rπ α 1 sin + α 3 sin + α 5 sin + α 7 sin r r (19) Ou of all he independen combinaions of α 1, α 3, α 5 and α 7, wo give BBP-ype series for K. The BBP-ype formulas for K are found under [α 1, α 3, α 5, α 7 ] = [1, 1, 1, 1] and [α 1, α 3, α 5, α 7 ] = [1, 1, 1, 1]. 9.1 Case [α 1, α 3, α 5, α 7 ] = [1, 1, 1, 1] Replicaes Eq. () 9.2 Case [α 1, α 3, α 5, α 7 ] = [1, 1, 1, 1] Replicaes Eq. (4) 10 n = 10: Base 10 formulas Choosing n = 10 in Eq. (2), we find 1 iπ 1 3iπ K = α 1 Im Li 1 exp + α 3 Im Li 1 exp iπ 1 9iπ + α 7 Im Li 1 exp + α 9 Im Li 1 exp ) ( ) ( ) ( )]} 3rπ 7rπ 9rπ α 1 sin + α 3 sin + α 7 sin + α 9 sin r r () 39

8 Ou of all he independen combinaions of α 1, α 3, α 7 and α 9, wo give BBP-ype series for K. The BBP-ype formulas for K are found under [α 1, α 3, α 7, α 9 ] = [1, 1, 1, 1] and [α 1, α 3, α 7, α 9 ] = [1, 1, 1, 1] Case [α 1, α 3, α 7, α 9 ] = [1, 1, 1, 1] Using α 1 = 1 = α 3 = α 7 = α 9 in Eq. () we find ( arcan + (δ δ 2 )π = 5 P (1, 19 10,, ( 9, 0, 8, 0, 0, 0, 6, 0, 5, 0, (21) 10.2 Case [α 1, α 3, α 7, α 9 ] = [1, 1, 1, 1] 4, 0, 3, 0, 0, 0,, 0, 1, 0)) Using α 1 = 1 = α 3 = α 7 = α 9 in Eq. () we find ( 1 arcan = 1 P (1, 19 10,, ( 9, 0, 8, 0, 4 7, 0, 6, 0, (22) 5, 0, 4, 0, 3, 0, 4 2, 0,, 0, 1, 0)) 11 n = 12: Bases 12 and 2 12 formulas Choosing n = 12 in Eq. (2), we find 1 iπ 1 5iπ K = α 1 Im Li 1 exp + α 5 Im Li 1 exp iπ 1 11iπ + α 7 Im Li 1 exp + α 11 Im Li 1 exp ) ( ) ( ) ( )]} 5rπ 7rπ 11rπ α 1 sin + α 5 sin + α 7 sin + α 11 sin r r Ou of all he independen combinaions of α j, j = 1, 5, 7, 11, BBP-ype series exis for he following α j combinaions: [1, 1, 1, 1] [1, 1, 1, 1] [1, 1, 1, 1] [1, 1, 1, 1] [α 1, α 5, α 7, α 11 ] = [1, 0, 1, 0] [1, 0, 1, 0] [0, 1, 0, 1] [0, 1, 0, 1] (23) ()

9 11.1 Case [α 1, α 5, α 7, α 11 ] = [1, 1, 1, 1] ( arcan + π 6 (δ δ 2 ) = 6 P (1, 23 12,, ( 11, 0, 0, 0, 9, 0, 8, (25) 0, 0, 0, 6, 0, 5, 0, 0, 0, 3, 0, 2, 0, 0, 0, 1, 0)) 11.2 Case [α 1, α 5, α 7, α 11 ] = [1, 1, 1, 1] Replicaes Eq. (18) 11.3 Case [α 1, α 5, α 7, α 11 ] = [1, 1, 1, 1] Replicaes Eq. (8) Case [α 1, α 5, α 7, α 11 ] = [1, 1, 1, 1] ( 2 2 arcan = 2 23 P (1, 12,, ( 11, 0, 2 10, 0, 9, 0, 8, 0, (26) 2 7, 0, 6, 0, 5, 0, 2 4, 0, 3, 0, 2, 0, 2, 0, 1, 0)) 11.5 Case [α 1, α 5, α 7, α 11 ] = [1, 0, 1, 0] ( 3 3 arcan = 96 P (1, 23 96,, (48 22, 0, 0, 10 19, , 0, , 256, 0, 0, 64 12, 0, 32 10, 0, 0, 16 7, 8 6, 0, 4 4, 4 3, 0, 0, 1, 0)) 11.6 Case [α 1, α 5, α 7, α 11 ] = [1, 0, 1, 0] 1 arcan = ( P (1, 96,, (48 22, 48 21, 48, 0, , 10 17, , 0, , , 64 12, 0, 32 10, 32 9, 32 8, 0, 8 6, 16 5, 4 4, 0, 4 2, 2, 1, 0)) (27) (28) 41

10 11.7 Case [α 1, α 5, α 7, α 11 ] = [0, 1, 0, 1] ( 3 3 arcan = 96 P (1, 23 96,, (48 22, 0, 0, 10 19, , 0, , 256, 0, 0, 64 12, 0, 32 10, 0, 0, 16 7, 8 6, 0, 4 4, 4 3, 0, 0, 1, 0)) (29) 11.8 Case [α 1, α 5, α 7, α 11 ] = [0, 1, 0, 1] ( arcan = 96 P (1, 23 96,, (48 22, 48 21, 48, 0, , 10 17, , 0, , , 64 12, 0, 32 10, 32 9, 32 8, 0, 8 6, 16 5, 4 4, 0, 4 2, 2, 1, 0)) We remark here ha he binary BBP formulas ha resul by evaluaing a = 2 p 1, p Z + Eqs. (27), (28), (29) and (), have been discussed elsewhere (see [7]). () 12 n = : Bases and formulas Choosing n = in Eq. (2), we find 1 iπ 1 2iπ K = α 1 Im Li 1 exp + α 2 Im Li 1 exp 1 4iπ 1 7iπ + α 4 Im Li 1 exp + α 7 Im Li 1 exp 1 8iπ 1 11iπ + α 8 Im Li 1 exp + α 11 Im Li 1 exp 1 13iπ 1 14iπ + α 13 Im Li 1 exp + α 14 Im Li 1 exp ) ( ) ( ) 2rπ 4rπ α 1 sin + α 2 sin + α 4 sin r r ( ) ( ) ( ) 7rπ 8rπ 11rπ + α 7 sin + α 8 sin + α 11 sin ( ) ( )]} 13rπ 14rπ + α 13 sin + α 14 sin (31) 42

11 BBP-ype series exis for he following α j combinaions: [1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1] [α 1, α 2, α 4, α 7, α 8, α 11, α 13, α 14 ] = [1, 0, 0, 1, 0, 1, 1, 0] [1, 0, 0, 1, 0, 1, 1, 0] [0, 1, 1, 0, 1, 0, 0, 1] [0, 1, 1, 0, 1, 0, 0, 1] (32) 12.1 Case [α 1, α 2, α 4, α 7, α 8, α 11, α 13, α 14 ] = [1, 1, 1, 1, 1, 1, 1, 1] ( arcan + π = 29 P (1,,, ( 14, 0, 0, 0, 0, 0, 11, 0, 0, 0, 6 j=1 δ j (33) 9, 0, 8, 0, 0, 0, 6, 0, 5, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0)) 12.2 Case [α 1, α 2, α 4, α 7, α 8, α 11, α 13, α 14 ] = [1, 1, 1, 1, 1, 1, 1, 1] 2 arcan ( π δ 1 = 14 P (1,,, ( 13, 12, 0, 10, 0, 0, 7, 6, 0, 0, 3, 0,, 1, 0)) 12.3 Case [α 1, α 2, α 4, α 7, α 8, α 11, α 13, α 14 ] = [1, 1, 1, 1, 1, 1, 1, 1] ( arcan + π 3 δ = 3 P (1, 29,, ( 14, 0, 0, 0, 4 12, 0, 11, 0, 0, 0, (34) (35) 9, 0, 8, 0, 0, 0, 6, 0, 5, 0, 0, 0, 3, 0, 4 2, 0, 0, 0, 1, 0)) 12.4 Case [α 1, α 2, α 4, α 7, α 8, α 11, α 13, α 14 ] = [1, 1, 1, 1, 1, 1, 1, 1] 2 ( arcan = 3 P (1, 14,, ( 13, 12, 0, 10, 4 9, 0, 7, 6, 0, 4 4, 3, 0,, 1, 0)) (36) 43

12 12.5 Case [α 1, α 2, α 4, α 7, α 8, α 11, α 13, α 14 ] = [1, 0, 0, 1, 0, 1, 1, 0] 2 ( arcan + 2π 3 δ = 3 P (1, 29,, ( 28, 27, 0, 25, 4, 0, 22, 21, 0, 4 19, 18, 0, 16,, 0, 13, 12, 0, 10, 4 9, 0, 7, 6, 0, 4 4, 3, 0,, 1, 0)) 12.6 Case [α 1, α 2, α 4, α 7, α 8, α 11, α 13, α 14 ] = [1, 0, 0, 1, 0, 1, 1, 0] 2 arcan ( = 29 P (1,,, ( 28, 27, 0, 25, 0, 0, 22, 21, 0, 0, 18, 0, 16,, 0, 13, 12, 0, 10, 0, 0, 7, 6, 0, 0, 3, 0,, 1, 0)) 12.7 Case [α 1, α 2, α 4, α 7, α 8, α 11, α 13, α 14 ] = [0, 1, 1, 0, 1, 0, 0, 1] 2 arcan ( = 29 P (1,,, ( 28, 27, 0, 25, 0, 0, 22, 21, 0, 0, 18, 0, 16,, 0, 13, + 2π δ 1 12, 0, 10, 0, 0, 7, 6, 0, 0, 3, 0,, 1, 0)) 12.8 Case [α 1, α 2, α 4, α 7, α 8, α 11, α 13, α 14 ] = [0, 1, 1, 0, 1, 0, 0, 1] 2 ( arcan = 3 P (1, 29,, ( 28, 27, 0, 25, 4, 0, 22, 21, 0, 4 19, 18, 0, 16,, 0, 13, 12, 0, 10, 4 9, 0, 7, 6, 0, 4 4, 3, 0,, 1, 0)) Noe ha Eq. () is an expanded version of Eq. (36). (37) (38) (39) () 13 n = : Bases and 2 formulas Choosing n = in Eq. (2), we find 44

13 1 iπ 1 3iπ K = α 1 Im Li 1 exp + α 3 Im Li 1 exp 1 7iπ 1 9iπ + α 7 Im Li 1 exp + α 9 Im Li 1 exp 1 11iπ 1 13iπ + α 11 Im Li 1 exp + α 13 Im Li 1 exp 1 17iπ 1 19iπ + α 17 Im Li 1 exp + α 19 Im Li 1 exp ) ( ) ( ) 3rπ 7rπ α 1 sin + α 3 sin + α 7 sin r r ( ) ( ) ( ) 9rπ 11rπ 13rπ + α 9 sin + α 11 sin + α 13 sin ( ) ( )]} 17rπ 19rπ + α 17 sin + α 19 sin BBP-ype series exis for he following α j combinaions: [1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1] [α 1, α 3, α 7, α 9, α 11, α 13, α 17, α 19 ] = [1, 0, 1, 1, 0, 0, 1, 0] [1, 0, 1, 1, 0, 0, 1, 0] [0, 1, 0, 0, 1, 1, 0, 1] [0, 1, 0, 0, 1, 1, 0, 1] (41) (42) 13.1 Case [α 1, α 3, α 7, α 9, α 11, α 13, α 17, α 19 ] = [1, 1, 1, 1, 1, 1, 1, 1] ( arcan + π 2 δ = 2 39 P (1,,, ( 19, 0, 18, 0, 4 17, 0, 16, 0,, 0, 14, 0, 13, 0, 4 12, 0, 11, (43) 0, 10, 0, 9, 0, 8, 0, 4 7, 0, 6, 0, 5, 0, 4, 0, 3, 0, 4 2, 0,, 0, 1, 0)) 13.2 Case [α 1, α 3, α 7, α 9, α 11, α 13, α 17, α 19 ] = [1, 1, 1, 1, 1, 1, 1, 1] ( 10 5 arcan (δ 1 + δ 2 + δ 3 )π 5 = P (1,,, ( 19, 0, 18, 0, 0, 0, 16, 0,, 0, 14, 0, 13, 0, 0, 0, 11, 0, 10, 0, (44) 9, 0, 8, 0, 0, 0, 6, 0, 5, 0, 4, 0, 3, 0, 0, 0,, 0, 1, 0)) 45

14 13.3 Case [α 1, α 3, α 7, α 9, α 11, α 13, α 17, α 19 ] = [1, 1, 1, 1, 1, 1, 1, 1] 5 arcan ( π 5 δ = 5 (45) P (1, 19,, ( 18, 0, 16, 0, 0, 0, 12, 0, 10, 0, 8, 0, 6, 0, 0, 0, 2, 0, 1, 0)) Noe ha Eq. (45) and Eq. (21) are idenical. Addiion of Eqs. (44) and (45), boh evaluaed a = 2, gives a very ineresing binary BBP-ype formula for π 5: π 5 = P (1, 2,, (2 19, 2 19, 2 18, 0, 0, 2 17, 2 16, 0, 2, 0, 2 14, 0, 2 13, 2 13, 0, 0, 2 11, 2 11, 2 10, 0, 2 9, 2 9, 2 8, 0, 0, 2 7, 2 6, 0, 2 5, 0, 2 4, 0, 2 3, 2 3, 0, 0, 2, 2, 1, 0)) 13.4 Case [α 1, α 3, α 7, α 9, α 11, α 13, α 17, α 19 ] = [1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (22) (46) 13.5 Case [α 1, α 3, α 7, α 9, α 11, α 13, α 17, α 19 ] = [1, 0, 1, 1, 0, 0, 1, 0] 5 arcan 5 = ( P (1, ,, ( , , , 0, 0, , , 0, 32768, 0, , 0, , , 0, 0, 48 22, 48 21, 10, 0, , , , 0, 0, , 64 12, 0, 32 10, 0, 16 8, 0, 8 6, 8 5, 0, 0, 2 2, 2, 1, 0)) Noe ha a = 1, Eq. (46) is recovered Case [α 1, α 3, α 7, α 9, α 11, α 13, α 17, α 19 ] = [1, 0, 1, 1, 0, 0, 1, 0] arcan = ( P (1, ,, ( , , , 0, , , , 0, 32768, , , 0, , , 16384, 0, 48 22, 48 21, 10, 0, , , , 0, , , 64 12, 0, 32 10, 128 9, 16 8, 0, 8 6, 8 5, 16 4, 0, 2 2, 2, 1, 0)) (47) (48) 46

15 13.7 Case [α 1, α 3, α 7, α 9, α 11, α 13, α 17, α 19 ] = [0, 1, 0, 0, 1, 1, 0, 1] arcan = ( P (1, ,, ( , , , 0, , , , 0, 32768, , , 0, , , 16384, 0, 48 22, 48 21, 10, 0, , , , 0, , , 64 12, 0, 32 10, 128 9, 16 8, 0, 8 6, 8 5, 16 4, 0, 2 2, 2, 1, 0)) 13.8 Case [α 1, α 3, α 7, α 9, α 11, α 13, α 17, α 19 ] = [0, 1, 0, 0, 1, 1, 0, 1] 5 arcan 5 = ( P (1, ,, ( , , , 0, 0, , , 0, 32768, 0, , 0, , , 0, 0, 48 22, 48 21, 10, 0, , , , 0, 0, , 64 12, 0, 32 10, 0, 16 8, 0, 8 6, 8 5, 0, 0, 2 2, 2, 1, 0)) (49) (50) 14 n = : Base formulas Choosing n = in Eq. (2), we find 1 iπ 1 5iπ K = α 1 Im Li 1 exp + α 5 Im Li 1 exp 1 7iπ 1 11iπ + α 7 Im Li 1 exp α 11 Im Li 1 exp 1 13iπ 1 17iπ + α 13 Im Li 1 exp + α 17 Im Li 1 exp 1 19iπ 1 23iπ + α 19 Im Li 1 exp + α 23 Im Li 1 exp (51) ) ( ) ( ) 5rπ 7rπ α 1 sin + α 5 sin + α 7 sin r r ( ) ( ) ( ) 11rπ 13rπ 17rπ + α 11 sin + α 13 sin + α 17 sin ( ) ( )]} 19rπ 23rπ +α 19 sin + α 23 sin 47

16 BBP-ype series exis for he following α j combinaions: [1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1] [α 1, α 5, α 7, α 11, α 13, α 17, α 19, α 23 ] = [1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1] (52) 14.1 Case [α 1, α 5, α 7, α 11, α 13, α 17, α 19, α 23 ] = [1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (25) 14.2 Case [α 1, α 5, α 7, α 11, α 13, α 17, α 19, α 23 ] = [1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (18) 14.3 Case [α 1, α 5, α 7, α 11, α 13, α 17, α 19, α 23 ] = [1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (8) 14.4 Case [α 1, α 5, α 7, α 11, α 13, α 17, α 19, α 23 ] = [1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (26) n = : Base formulas Choosing n = in Eq. (2), we find 1 iπ 1 7iπ K = α 1 Im Li 1 exp + α 7 Im Li 1 exp 1 11iπ 1 13iπ + α 11 Im Li 1 exp α 13 Im Li 1 exp 1 17iπ 1 19iπ + α 17 Im Li 1 exp + α 19 Im Li 1 exp 1 23iπ 1 29iπ + α 23 Im Li 1 exp + α 29 Im Li 1 exp (53) ) ( ) ( ) 7rπ 11rπ α 1 sin + α 7 sin + α 11 sin r r ( ) ( ) ( ) 13rπ 17rπ 19rπ + α 13 sin + α 17 sin + α 19 sin ( ) ( )]} 23rπ 29rπ +α 23 sin + α 29 sin 48

17 BBP-ype series exis for he following α j combinaions: [1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1] [α 1, α 7, α 11, α 13, α 17, α 19, α 23, α 29 ] = [1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1] (54).1 Case [α 1, α 7, α 11, α 13, α 17, α 19, α 23, α 29 ] = [1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (38).2 Case [α 1, α 7, α 11, α 13, α 17, α 19, α 23, α 29 ] = [1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (37).3 Case [α 1, α 7, α 11, α 13, α 17, α 19, α 23, α 29 ] = [1, 1, 1, 1, 1, 1, 1, 1] ( arcan = 5 P (1, 59,, ( 29, 0, 2 28, 0, 0, 0, 26, 0, 2 25, 0,, 0, 23, 0, 0, 0, 21, 0,, 0, 2 19, 0, 18, 0, 0, 0, 2 16, 0,, 0, 14, 0, 2 13, 0, 0, 0, 11, 0, 2 10, 0, 9, 0, 8, 0, 0, 0, 6, 0, 5, 0, 2 4, 0, 3, 0, 0, 0, 2, 0, 1, 0)).4 Case [α 1, α 7, α 11, α 13, α 17, α 19, α 23, α 29 ] = [1, 1, 1, 1, 1, 1, 1, 1] arcan ( = 1 59 P (1,,, ( 29, 0, 2 28, 0, 4 27, 0, 26, 0, 2 25, 0,, 0, 23, 0, 8 22, 0, 21, 0,, 0, 2 19, 0, 18, 0, 4 17, 0, 2 16, 0,, 0, 14, 0, 2 13, 0, 4 12, 0, 11, 0, 2 10, 0, 9, 0, 8, 0, 8 7, 0, 6, 0, 5, 0, 2 4, 0, 3, 0, 4 2, 0, 2, 0, 1, 0)) (55) (56) 49

18 16 n = : Base formulas Choosing n = in Eq. (2), we find 1 iπ 1 3iπ K = α 1 Im Li 1 exp + α 3 Im Li 1 exp 1 7iπ 1 9iπ + α 7 Im Li 1 exp + α 9 Im Li 1 exp 1 11iπ 1 13iπ + α 11 Im Li 1 exp + α 13 Im Li 1 exp 1 17iπ 1 19iπ + α 17 Im Li 1 exp + α 19 Im Li 1 exp 1 21iπ 1 23iπ + α 21 Im Li 1 exp + α 23 Im Li 1 exp 1 27iπ 1 29iπ + α 27 Im Li 1 exp + α 29 Im Li 1 exp 1 31iπ 1 33iπ + α 31 Im Li 1 exp + α 33 Im Li 1 exp 1 37iπ 1 39iπ + α 37 Im Li 1 exp + α 39 Im Li 1 exp (57) ) ( ) ( ) 3rπ 7rπ α 1 sin + α 3 sin + α 7 sin r r ( ) ( ) ( ) ( ) 9rπ 11rπ 13rπ 17rπ + α 9 sin + α 11 sin + α 13 sin + α 17 sin ( ) ( ) ( ) ( ) 19rπ 21rπ 23rπ 27rπ + α 19 sin + α 21 sin + α 23 sin + α 27 sin ( ) ( ) ( ) 29rπ 31rπ 33rπ + α 29 sin + α 31 sin + α 33 sin ( ) ( )]} 37rπ 39rπ + α 37 sin + α 39 sin BBP-ype series exis for he following α j combinaions: [α 1, α 3, α 7, α 9, α 11, α 13, α 17, α 19, α 21, α 23, α 27, α 29, α 31, α 33, α 37, α 39 ] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (58) 16.1 Case [α 1, α 3, α 7, α 9, α 11, α 13, α 17, α 19, α 21, α 23, α 27, α 29, α 31, α 33, α 37, α 39 ] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (21) 50

19 16.2 Case [α 1, α 3, α 7, α 9, α 11, α 13, α 17, α 19, α 21, α 23, α 27, α 29, α 31, α 33, α 37, α 39 ] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (43) 16.3 Case [α 1, α 3, α 7, α 9, α 11, α 13, α 17, α 19, α 21, α 23, α 27, α 29, α 31, α 33, α 37, α 39 ] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (44) 16.4 Case [α 1, α 3, α 7, α 9, α 11, α 13, α 17, α 19, α 21, α 23, α 27, α 29, α 31, α 33, α 37, α 39 ] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (22) 17 n = : Base formulas Choosing n = in Eq. (2), we find 1 iπ 1 7iπ K = α 1 Im Li 1 exp + α 7 Im Li 1 exp 1 11iπ 1 13iπ + α 11 Im Li 1 exp + α 13 Im Li 1 exp 1 17iπ 1 19iπ + α 17 Im Li 1 exp + α 19 Im Li 1 exp 1 23iπ 1 29iπ + α 23 Im Li 1 exp + α 29 Im Li 1 exp 1 31iπ 1 37iπ + α 31 Im Li 1 exp + α 37 Im Li 1 exp 1 41iπ 1 43iπ + α 41 Im Li 1 exp + α 43 Im Li 1 exp 1 47iπ 1 49iπ + α 47 Im Li 1 exp + α 49 Im Li 1 exp 1 53iπ 1 59iπ + α 53 Im Li 1 exp + α 59 Im Li 1 exp (59) ) ( ) ( ) 7rπ 11rπ α 1 sin + α 7 sin + α 11 sin r r ( ) ( ) ( ) ( ) 13rπ 17rπ 19rπ 23rπ + α 13 sin + α 17 sin + α 19 sin + α 23 sin ( ) ( ) ( ) ( ) 29rπ 31rπ 37rπ 41rπ + α 29 sin + α 31 sin + α 37 sin + α 41 sin ( ) ( ) ( ) 43rπ 47rπ 49rπ + α 43 sin + α 47 sin + α 49 sin ( ) ( )]} 53rπ 59rπ + α 53 sin + α 59 sin 51

20 BBP-ype series exis for he following α j combinaions: [α 1, α 7, α 11, α 13, α 17, α 19, α 23, α 29, α 31, α 37, α 41, α 43, α 47, α 49, α 53, α 59 ] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] () 17.1 Case [α 1, α 7, α 11, α 13, α 17, α 19, α 23, α 29, α 31, α 37, α 41, α 43, α 47, α 49, α 53, α 59 ] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] ( arcan π 5 5 δ j + 2π 5δ 1 j=2 = P (1,, 1, ( 59, 0, 2 58, 0, 0, 0, 56, 0, 2 55, 0, 54, 0, 53, 0, 0, 0, 51, 0, 50, 0, 2 49, 0, 48, 0, 0, 0, 2 46, 0, 45, 0, 44, 0, 2 43, 0, 0, 0, 41, 0, 2, 0, 39, 0, 38, 0, 0, 0, 36, 0, 35, 0, 2 34, 0, 33, 0, 0, 0, 2 31, 0,, 0, 29, 0, 2 28, 0, 0, 0, 26, 0, 2 25, 0,, 0, 23, 0, 0, 0, 21, 0,, 0, 2 19, 0, 18, 0, 0, 0, 2 16, 0,, 0, 14, 0, 2 13, 0, 0, 0, 11, 0, 2 10, 0, 9, 0, 8, 0, 0, 0, 6, 0, 5, 0, 2 4, 0, 3, 0, 0, 0, 2, 0, 1, 0)) (61) 17.2 Case [α 1, α 7, α 11, α 13, α 17, α 19, α 23, α 29, α 31, α 37, α 41, α 43, α 47, α 49, α 53, α 59 ] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] 3 arcan 6 ( = P (1,, 1, ( 59, 0, 0, 0, 4 57, 0, 56, 0, 0, 0, 54, 0, 53, 0, 0, 0, 51, 0, 50, 0, 0, 0, 48, 0, 4 47, 0, 0, 0, 45, 0, 44, 0, 0, 0, 4 42, 0, 41, 0, 0, 0, 39, 0, 38, 0, 0, 0, 36, 0, 35, 0, 0, 0, 33, 0, 4 32, 0, 0, 0,, 0, 29, 0, 0, 0, 4 27, 0, 26, 0, 0, 0,, 0, 23, 0, 0, 0, 21, 0,, 0, 0, 0, 18, 0, 4 17, 0, 0, 0,, 0, 14, 0, 0, 0, 4 12, 0, 11, 0, 0, 0, 9, 0, 8, 0, 0, 0, 6, 0, 5, 0, 0, 0, 3, 0, 4 2, 0, 0, 0, 1, 0)) (62) 52

21 17.3 Case [α 1, α 7, α 11, α 13, α 17, α 19, α 23, α 29, α 31, α 37, α 41, α 43, α 47, α 49, α 53, α 59 ] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (37) 17.4 Case [α 1, α 7, α 11, α 13, α 17, α 19, α 23, α 29, α 31, α 37, α 41, α 43, α 47, α 49, α 53, α 59 ] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (55) 17.5 Case [α 1, α 7, α 11, α 13, α 17, α 19, α 23, α 29, α 31, α 37, α 41, α 43, α 47, α 49, α 53, α 59 ] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] ( arcan π 12 δ j + 2π δ 1 j=2 = P (1,, 1, ( 59, 0, 0, 0, 0, 0, 56, 0, 0, 0, 54, 0, 53, 0, 0, 0, 51, 0, 50, 0, 0, 0, 48, 0, 0, 0, 0, 0, 45, 0, 44, 0, 0, 0, 0, 0, 41, 0, 0, 0, 39, 0, 38, 0, 0, 0, 36, 0, 35, 0, 0, 0, 33, 0, 0, 0, 0, 0,, 0, 29, 0, 0, 0, 0, 0, 26, 0, 0, 0,, 0, 23, 0, 0, 0, 21, 0,, 0, 0, 0, 18, 0, 0, 0, 0, 0,, 0, 14, 0, 0, 0, 0, 0, 11, 0, 0, 0, 9, 0, 8, 0, 0, 0, 6, 0, 5, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0)) (63) 17.6 Case [α 1, α 7, α 11, α 13, α 17, α 19, α 23, α 29, α 31, α 37, α 41, α 43, α 47, α 49, α 53, α 59 ] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] ( arcan = P (1, 119, 1, ( 59, 0, 2 58, 0, 4 57, 0, 56, 0, 2 55, 0, 54, 0, 53, 0, 8 52, 0, 51, 0, 50, 0, 2 49, 0, 48, 0, 4 47, 0, 2 46, 0, 45, 0, 44, 0, 2 43, 0, 4 42, 0, 41, 0, 2, 0, 39, 0, 38, 0, 8 37, 0, 36, 0, 35, 0, 2 34, 0, 33, 0, 4 32, 0, 2 31, 0,, 0, 29, 0, 2 28, 0, 4 27, 0, 26, 0, 2 25, 0,, 0, 23, 0, 8 22, 0, 21, 0,, 0, 2 19, 0, 18, 0, 4 17, 0, 2 16, 0,, 0, 14, 0, 2 13, 0, 4 12, 0, 11, 0, 2 10, 0, 9, 0, 8, 0, 8 7, 0, 6, 0, 5, 0, 2 4, 0, 3, 0, 4 2, 0, 2, 0, 1, 0)) (64) 17.7 Case [α 1, α 7, α 11, α 13, α 17, α 19, α 23, α 29, α 31, α 37, α 41, α 43, α 47, α 49, α 53, α 59 ] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (56) 53

22 17.8 Case [α 1, α 7, α 11, α 13, α 17, α 19, α 23, α 29, α 31, α 37, α 41, α 43, α 47, α 49, α 53, α 59 ] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] Replicaes Eq. (38). 18 Conclusion We have derived numerous new explici digi exracion BBP-ype formulas for he arcangens of real numbers in general bases,, wihou doing compuer searches. The high poin of his work is he discovery, for he firs ime, of a binary BBP-ype formula for π 5. References [1] Lord, N. Recen formulae for p: Arcan revisied! The Mahemaical Gazee, Vol. 83, 1999, [2] Bailey, D. H. A compendium of bbp-ype formulas for mahemaical consans, February 11. Available Online hp://crd.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf [3] Bailey, D. H., P. B. Borwein, S. Plouffe. On he rapid compuaion of various polylogarihmic consans. Mahemaics of Compuaion, Vol. 66, 1997, No. 218, [4] Bailey, D. H., R. E. Crandall. On he random characer of fundamenal consan expansions. Experimenal Mahemaics, Vol. 10, 01, p [5] Borwein, J. M., D. Borwein, W. F. Galway. Finding and excluding b-ary machin-ype BBP formulae, 02. Available Online hp://crd.lbl.gov/~dhbailey/dhbpapers/machin.pdf [6] Chamberland, M. Binary bbp-formulae for logarihms and generalized Gaussian Mersenne primes. Journal of Ineger Sequences, Vol. 6, 03. [7] Adegoke, K., J. O. Lafon, O. Layeni. A class of digi exracion BBP-ype formulas in general binary bases. Noes on Number Theory and Discree Mahemaics, Vol. 17, 11, No. 4,