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

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:

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]
Top of page