Home page Measures Eliminated sets Self-checking pairs "Tweaking"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

- 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.

Links to lists

- 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] Top of page