************** GREETINGS from PLOUFFE'S INVERTER ******************
Your request was processed as follow.
For each request with a DECIMAL number (please provide at least 16 digits).
167 tests are conducted, including :
141 using a smart lookup
+ 17 using generalized expansions
(Including the new Fibonacci expansion (called fibrep), in
which X = sum(1/fibonacci(i)), i is an increasing sequence.
+ 9 using Integer Relations Algorithms.
For each request with an INTEGER (please provide 5 digits minimum).
49 tests are conducted using a smart lookup.
For details see : http://www.lacim.uqam.ca/pi/server.html
IMPORTANT NOTE : All the tests are done IF the number of digits is 15
or MORE, with less digits results are NOT reliable.
ALL requests with 16 digits and more are recorded
and kept with the author name and mail address in case the
number would eventually become important. Some tests
will be conducted on those numbers and if ever an interesting
result is found the author would immediately be noticed
of the discovery. -Simon Plouffe
---------------------------------------------------------------------------
As of : Sat Nov 20 17:05:00 EST 1999 there are 92,803,842 mathematical constants in the database.
Report with K =
2.4142135623730950488
######################################################################
Report with Generalized Expansions : 16 different expansions
for details see : http://www.lacim.uqam.ca/pi/server.html
######################################################################
guessing a gen. fun. for continued fraction of x : , [2, 2, 2, 2, 2, 2, 2, 2, 2,
2
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], - -----
x - 1
egpfrac of x: FAIL : [2, 3, 13, 253, 218201, 61323543807]
egpprod of x: FAIL : [2, 3, 5, 5, 16, 18, 78, 102, 120, 144, 251, 363]
egpprod of 1/x: FAIL : [0, 3, 5, 5, 16, 18, 78, 102, 120, 144, 251, 363]
egpfrac of 1/x: FAIL : [0, 3, 13, 253, 218201, 61323543799]
altegpp of x: FAIL : [2, 2, 5, 7, 197, 199, 7761797, 10768170]
altegfr of x: FAIL : [2, 2, 11, 195, 180120, 120479426003]
altegpp of 1/x: FAIL : [0, 2, 5, 7, 197, 199, 7761798, 150825948]
altegfr of 1/x: FAIL : [0, 2, 11, 195, 180120, 120479425974]
factbas of x: FAIL : [2, 0, 0, 2, 1, 4, 4, 1, 5, 0, 8, 1, 11, 1, 7, 8, 4, 4,
4, 11, 13]
factbas of 1/x: FAIL : [0, 0, 0, 2, 1, 4, 4, 1, 5, 0, 8, 1, 11, 1, 7, 8, 4, 4
, 4, 11, 13]
binexp of x: FAIL : [2, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1,
1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1]
binexp of 1/x: FAIL : [0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1,
1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1]
fibrep of x : FAIL : [4, 7, 14, 16, 19, 23, 25, 28, 31, 35, 37, 39, 42, 46,
48, 55, 63, 66, 73, 75, 78, 81, 83, 87, 90, 93, 96, 100, 102, 105, 107, 111,
119]
fibrep of 1/x : FAIL : [4, 7, 14, 16, 19, 23, 25, 28, 31, 35, 37, 39, 42, 46,
48, 55, 63, 66, 73, 75, 78, 81, 83, 87, 90, 93, 96, 99, 102, 104, 106, 109, 111
, 117, 120]
######################################################################
Report with Integer Relation algorithms : 4 algebraic tests + 5 linear
combination tests.
######################################################################
2
K satisfies the following polynomial, -1 - 2 X + X
K does NOT satisfy a simple combination of the constants
[K, E, Pi, gamma, Ei(1), W(1), 1]
K does NOT satisfy a simple combination of the constants
[K, Pi*sqrt(3), log(3), log(2), gamma, Pi*sqrt(2)]
K does NOT satisfy a simple combination of the constants
[K, Pi**2, Catalan, Pi*log(2), Pi**2*sqrt(2), log(2)**2]
K does NOT satisfy a simple combination of the constants
[K, Pi**3, Zeta(3), Pi**2*log(2), log(2)**3, Pi**3*sqrt(3), Pi**3*sqrt(2)]
K satisfies the following Z-linear combination :
K [-1, 0, 0, 0, 1, 1, 0, 0]
with the constants, [K, Pi, exp(1), log(2), sqrt(2), 1, 1/Pi]
K does NOT satisfy a simple combination of the constants
[K, exp(Pi), exp(-Pi), exp(2*Pi), exp(-2*Pi), Pi, 1, 1/Pi]
Report with K =
2.4142135623730950488
######################################################################
Report with the smart lookup : elementary variations around K
Shows the first match of each operation if any.
######################################################################
Number Operation Table Expression
24142135623730950488 K*1 a007 (1+1*sqrt(2))/1
22130290988420037947 K*11/12 a007 (11+11*sqrt(2))/12
21727922061357855439 K*9/10 a007 (9+9*sqrt(2))/1
21459676109983067100 K*8/9 a007 (8+8*sqrt(2))/9
21124368670764581677 K*7/8 a007 (7+7*sqrt(2))/8
20693259106055100418 K*6/7 a007 (6+6*sqrt(2))/7
20118446353109125407 K*5/6 a007 (1+1*sqrt(2))/12
19313708498984760390 K*4/5 a007 (4+4*sqrt(2))/5
18777216596235183713 K*7/9 a007 (7+7*sqrt(2))/9
18106601717798212866 K*3/4 a007 (15+15*sqrt(2))/2
17244382588379250349 K*5/7 a007 (1+1*sqrt(2))/14
16899494936611665342 K*7/10 a007 (7+7*sqrt(2))/1
16094757082487300325 K*2/3 a007 (1+1*sqrt(2))/15
15088834764831844055 K*5/8 a007 (1+1*sqrt(2))/16
14485281374238570293 K*3/5 a007 (3+3*sqrt(2))/5
14082912447176387785 K*7/12 a007 (7+7*sqrt(2))/12
13795506070703400279 K*4/7 a007 (4+4*sqrt(2))/7
13412297568739416938 K*5/9 a007 (1+1*sqrt(2))/18
12071067811865475244 K*1/2 a007 (1+1*sqrt(2))/2
10729838054991533550 K*4/9 a007 (4+4*sqrt(2))/9
10346629553027550209 K*3/7 a007 (3+3*sqrt(2))/7
10059223176554562703 K*5/12 a007 (1+1*sqrt(2))/24
96568542494923801952 K*2/5 a007 (2+2*sqrt(2))/5
90533008588991064330 K*3/8 a007 (3+3*sqrt(2))/8
80473785412436501627 K*1/3 a007 (1+1*sqrt(2))/3
68977530353517001394 K*2/7 a007 (2+2*sqrt(2))/7
60355339059327376220 K*1/4 a007 sqrt((3+sqrt(8))/16)
53649190274957667751 K*2/9 a007 (2+2*sqrt(2))/9
40236892706218250813 K*1/6 a007 (1+1*sqrt(2))/6
34488765176758500697 K*1/7 a007 (1+1*sqrt(2))/7
30177669529663688110 K*1/8 a007 (1+1*sqrt(2))/8
26824595137478833876 K*1/9 a007 (1+1*sqrt(2))/9
20118446353109125407 K*1/12 a007 (1+1*sqrt(2))/12
34142135623730950488 K+1 a007 (2+1*sqrt(2))/1
33308802290397617155 K+11/12 a007 (23+12*sqrt(2))/12
33142135623730950488 K+9/10 a007 (19+10*sqrt(2))/1
33031024512619839377 K+8/9 a007 (17+9*sqrt(2))/9
32892135623730950488 K+7/8 a007 (15+8*sqrt(2))/8
32713564195159521917 K+6/7 a007 (13+7*sqrt(2))/7
32475468957064283821 K+5/6 a007 (11+6*sqrt(2))/6
32142135623730950488 K+4/5 a007 (18+10*sqrt(2))/1
31919913401508728266 K+7/9 a007 (16+9*sqrt(2))/9
31642135623730950488 K+3/4 a007 (7+4*sqrt(2))/4
31284992766588093345 K+5/7 a007 (12+7*sqrt(2))/7
31142135623730950488 K+7/10 a007 (17+10*sqrt(2))/1
30808802290397617155 K+2/3 a007 (5+3*sqrt(2))/3
30392135623730950488 K+5/8 a007 (13+8*sqrt(2))/8
30142135623730950488 K+3/5 a007 (16+10*sqrt(2))/1
29975468957064283821 K+7/12 a007 (19+12*sqrt(2))/12
29856421338016664774 K+4/7 a007 (11+7*sqrt(2))/7
29697691179286506044 K+5/9 a007 (14+9*sqrt(2))/9
29142135623730950488 K+1/2 a007 (15+10*sqrt(2))/1
28586580068175394932 K+4/9 a007 (13+9*sqrt(2))/9
28427849909445236202 K+3/7 a007 (10+7*sqrt(2))/7
28308802290397617155 K+5/12 a007 (17+12*sqrt(2))/12
28142135623730950488 K+2/5 a007 (14+10*sqrt(2))/1
27892135623730950488 K+3/8 a007 (11+8*sqrt(2))/8
27475468957064283821 K+1/3 a007 (4+3*sqrt(2))/3
26999278480873807631 K+2/7 a007 (9+7*sqrt(2))/7
26642135623730950488 K+1/4 a007 (5+4*sqrt(2))/4
26364357845953172710 K+2/9 a007 (11+9*sqrt(2))/9
25808802290397617155 K+1/6 a007 (7+6*sqrt(2))/6
25570707052302379059 K+1/7 a007 (8+7*sqrt(2))/7
25392135623730950488 K+1/8 a007 (9+8*sqrt(2))/8
25253246734842061599 K+1/9 a007 (10+9*sqrt(2))/9
24975468957064283821 K+1/12 a007 (13+12*sqrt(2))/12
14142135623730950488 K-1 a261 -4+2*x+2*x^2+3*x^3-2*x^5
14975468957064283821 K-11/12 a007 (1+12*sqrt(2))/12
15142135623730950488 K-9/10 a007 (1+10*sqrt(2))/1
15253246734842061599 K-8/9 a007 (1+9*sqrt(2))/9
15392135623730950488 K-7/8 a007 (1+8*sqrt(2))/8
15570707052302379059 K-6/7 a007 (1+7*sqrt(2))/7
15808802290397617155 K-5/6 a007 (1+6*sqrt(2))/6
16142135623730950488 K-4/5 a007 (1+5*sqrt(2))/5
16364357845953172710 K-7/9 a007 (2+9*sqrt(2))/9
16642135623730950488 K-3/4 a007 (1+4*sqrt(2))/4
16999278480873807631 K-5/7 a007 (2+7*sqrt(2))/7
17142135623730950488 K-7/10 a007 (3+10*sqrt(2))/1
17475468957064283821 K-2/3 a007 (1+3*sqrt(2))/3
17892135623730950488 K-5/8 a007 (3+8*sqrt(2))/8
18142135623730950488 K-3/5 a007 (2+5*sqrt(2))/5
18308802290397617155 K-7/12 a007 (5+12*sqrt(2))/12
18427849909445236202 K-4/7 a007 (3+7*sqrt(2))/7
18586580068175394932 K-5/9 a007 (4+9*sqrt(2))/9
19142135623730950488 K-1/2 a007 (1+2*sqrt(2))/2
19697691179286506044 K-4/9 a007 (5+9*sqrt(2))/9
19856421338016664774 K-3/7 a322 4/7+2^(1/2)
19975468957064283821 K-5/12 a323 7/12+2^(1/2)
20142135623730950488 K-2/5 a007 (6+10*sqrt(2))/1
20392135623730950488 K-3/8 a007 (5+8*sqrt(2))/8
20808802290397617155 K-1/3 a007 (2+3*sqrt(2))/3
21284992766588093345 K-2/7 a007 (5+7*sqrt(2))/7
21642135623730950488 K-1/4 a007 (15+20*sqrt(2))/2
21919913401508728266 K-2/9 a007 (7+9*sqrt(2))/9
22475468957064283821 K-1/6 a007 (5+6*sqrt(2))/6
22713564195159521917 K-1/7 a007 (6+7*sqrt(2))/7
22892135623730950488 K-1/8 a007 (7+8*sqrt(2))/8
23031024512619839377 K-1/9 a007 (8+9*sqrt(2))/9
23308802290397617155 K-1/12 a007 (11+12*sqrt(2))/12
48284271247461900976 2*K a007 (1+1*sqrt(2))/5
72426406871192851464 3*K a007 (3+3*sqrt(2))/1
16899494936611665342 7*K a007 (7+7*sqrt(2))/1
88137358701954302523 ln(K) m001 exp(-Pi)^GAM(1/12)-ln(1+sr(2))
12279471772995156799 ln(K+1) l277 log(roots(-2*x^2+4*x-1))
41132503787829275172 exp(K-1) h276 exp(roots(-2*x^3+2*x^2+4*x-4))
24311673443421421081 exp(1-K) h276 exp(roots(-2*x^3+2*x^2+4*x-4))
11180973760547916615 exp(K) h276 exp(roots(-2*x^2+4*x+2))
74691964546688150067 cos(K) l301 cos(tan(Pi*11/8))
66491431268670104129 sin(K) l302 sin(tan(Pi*11/8))
89437648403084672515e-1 exp(-K) h276 exp(roots(-2*x^2-4*x+2))
15537739740300373073 K^(1/2) a007 sqrt((1+sqrt(2))/1)
13415037626305777197 K^(1/3) m001 2^(1/3)^(ln(1+sr(2))/ln(2))
37511422009656572391 K^(3/2) a328 1/2*(10*2^(1/2)+14)^(1/2)*2^(1/2)
17996323451519974108 K^(2/3) m001 E^(2/3*ln(1+sr(2)))
64359425290558262475 1/K^(1/2) a007 sqrt((-1+sqrt(2))/1)
26658546821887205790 1/K^(3/2) a325 1/2*(10*2^(1/2)-14)^(1/2)*2^(1/2)
70710678118654752440 1/(1-K) a007 (0+1*sqrt(2))/2
29289321881345247560 1/(K+1) a007 sqrt((3-sqrt(8))/2)
11656854249492380195 (K+1)^2 a007 (3+2*sqrt(2))/5
41421356237309504880 (K-1)/(K+1) a007 sqrt((3-sqrt(8))/1)
24142135623730950488 (K+1)/(K-1) a007 (1+1*sqrt(2))/1
5857864376269049512 trunc(K)+1-K a007 (2-1*sqrt(2))/1
41421356237309504880 1/K a007 sqrt((3-sqrt(8))/1)
58284271247461900976 K^2 a007 (3+2*sqrt(2))/1
17157287525380990240 1/K^2 a007 (3-2*sqrt(2))/1
14071067811865475244 K^3 a007 (7+5*sqrt(2))/1
14142135623730950488 abs(K-1) a261 -4+2*x+2*x^2+3*x^3-2*x^5
4142135623730950488 K-2 a007 sqrt((3-sqrt(8))/1)
5857864376269049512 K-3 a007 (2-1*sqrt(2))/1
34142135623730950488 K+1 a007 (2+1*sqrt(2))/1
44142135623730950488 K+2 a007 (3+1*sqrt(2))/1
58578643762690495120 1-1/K a007 (2-1*sqrt(2))/1
76846804426234370630 K/Pi m192 (1+2^(1/2))/Pi
11780972450961724644 arctan(K) p199 S(1,1)[1,2,2,1,-1,1,-2,0]
75844755917481594856 Pi*K g141 Psi(1/8)-Psi(7/8)
Inverseur de Plouffe
Plouffe's Inverter
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Copyright (c) 1986-2000, Simon Plouffe
