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SELECT REQUIRED LIST FROM THIS LINE OR NEXT LINE: 2 & 3 term 4-term
(D=) 2 3 4 5 6 7 8 9 10 11 12 13 14
IDENTITY LISTS - CLASSIFICATION,
LAYOUT and LINKS
N.B.:
THE NOTATION "[x]" IS USED TO REPRESENT "arccot(x)"
(= arctan(1/x)) THROUGHOUT THIS SITE.
- The identities are listed on
separate pages, in the following categories and sub-categories:
- Identies having a specified
number (T) of terms (T=2, 3 or 4): 2-term and 3-term identities
are combined here.
- Identities with more than 4
terms ("half integers" counting as 2 terms), categorised
according to the least number of decimal digits, D, in any of
the cotangent values (integers, or in the case of half-integers,
their numerators) present in those terms - there are 13 sub-categories
of these (see "D=" links
above), for values of D from 2 to 14. Thus, for example, identities
in the D=7 sub-category are referred to as "7-digit identities"
(though they are almost certain to include terms having cotangent
values with more than 7 digits).
Identities in all lists are credited,
wherever possible, to the original discoverer - the initials MRW
or HCL are used when one of the editors claims credit. The initials
ASN refer to contributions from Amrik Singh Nimbran.
Within each category or sub-category,
identities appear in order of increasing "measure".
Each "D=" sub-category includes the "best 20"
identities (according to measure) for that value of D; others
may appear, if they have "historic" interest, or are
the originals of "tweaked"
identities, or are members of "self-checking
pairs", etc. The entries in the 2-, 3- and 4-term categories
are virtually all "historic".
Practical experience has shown
that the number of terms T in an identity ("half integers"
counting as one term) is likely to be related to D as follows:
D 3 4 5 6 7 8 9 10 11 12 13 14
T <7 7-8 9-10 11-12 13-14 15-16 17-18 19-21 22-23 24-26 27-28 29-31
Obviously, there
are exceptions to this rule - e.g. the 12- and 15- (integer) term
7-digit identities with measures 1.73115 and 1.70687, respectively;
and the 13-term 6-digit identity with measure 1.65738.
The "best" (lowest
measure) identities are to be found in the (D=) 3-digit and 4-digit
sub-categories.
Links to lists
LAYOUT OF IDENTITIES
- The first line of each identity
gives its measure, discoverer,
date of discovery (if known), and any cross-references to the
measures of related identities - with the symbol '¢¢'
if referring to the compound measure of a mutually-checking pair
of which the identity is a member, or an asterisk if another
identity exists from which the present one can be derived by
"tweaking" (it does not imply that the discoverer followed
that route!). In the latter case, the identity referred to may
be a "back-formation" created by the editors in order
to provide an "untweaked" prototype, with a valid "eliminated
set", from which the referring identity could have been
derived by "tweaking" - this is usually done to facilitate
checks of the validity of the referring identity. In such cases,
the discovery date of the "back-formed" identity is
shown as "(***)".
The second and subsequent lines contain the RHS terms of the
identity (the LHS is only present if it is a non-unit multiple
of [1] - otherwise PI/4 is assumed). The
"eliminated set" (if present) ends the entry (refer
to explanation).
Examples:
1.26579 MRW (22Sep04) (*1.31102, ¢¢2.11606,¢¢2.23061)
83[107]+17[1710]-22[103697]-12[2513489/2]-22[18280007883/2]
1.61934 HCL (03Jul03) (*1.66050)
2[1]=36625[39356]+25699[51412]-1699[96704]+21398[362619/2]+27911[734557]
-14241[6826318]+30649[7626068]-1333[21072618]+1699[46758414789]
+20236[2539791558]
Eliminated:{B}[11][23][[37][82][109]([96704])
2.99273 STØRMER (1895)
4[5]-[70]+[100]+[5000]-[10101]
Eliminated:[2&12]([5])[27&37]
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