Updates Reduction process Software PI evaluation programs Measures Tweaking LOGOUT / REFERENCESIdentity lists (Classification, Layout, & Links)Self-checking pairs of identities

The Machination software we have used to generate our own contribution to these lists depends on the beautiful "reduction" process devised by Professor John Todd (1911-2007). In his 1949 paper "A problem on arc tangent relations", Todd showed how the inverse cotangents of certain integers are "irreducible", all these "irreducibles" (apart from [1]) being associated in a 1-1 correspondence with the primes congruent to 1 (modulo 4) - for example, because n=9 is the smallest integer for which 41 divides (n² + 1), [9] is irreducible and is associated with the prime 41. Accordingly, [1] is associated with the prime 2; the irreducibles associated with the primes 5, 13, 17, 29 and 37 are respectively [2], [5], [4], [12] and [6], and so on (for further details, refer to "Reduction process"). Like "prime", the word "irreducible" may be used as a noun or an adjective, according to context.

Just as any composite integer can be expressed uniquely as a product of primes, Todd's process enables any inverse integral cotangent which is not itself irreducible to be expressed as a unique linear sum of irreducibles with positive or negative integer coefficients, i.e. "reduced" (in any such "reduction" the irreducible cotangent values are all numerically less than the reduced integer cotangent value). The process can also be used to obtain reductions of inverse rational cotangents, in particular of the inverse cotangents of half-integer values. By convention, the terms in any reduction are arranged in increasing sequence of irreducible cotangent values.

If the largest prime factor of (n² + 1) is less than 2n, [n] is not irreducible, so [3], [7], [8], [13], etc., are not irreducibles; their "reductions" are:

[3] = [1] - [2][7] = -[1] + 2[2][8] = [1] - [2] - [5][13] = [1] - [2] - [4]etc.

Measuring arccotangents in radians,
[1] = PI/4; eliminating [2] from the above reductions of [3] and
[7], the equation ([1]=) PI/4 = 2[3] + [7] is obtained. The discovery
of this **identity** (the term used herein to refer to such
arccotangent 'formulae' for PI/4) is nowadays credited to John
Machin (1680-1751). The reduction of [239] is -[1]+4[5], giving
([1]=) PI/4 = 4[5] - [239], the exact equivalent of Machin's own
celebrated identity of 1706.

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Our software operates on a database, continually being extended, of reductions of both integer and "half-integer" arccotangents, an example of the latter being: [1011/2] = [1]-3[2]+[4]+[5]+[6]. Half-integer terms of the form "[N/2]" appear explicitly in our identities - in practice, they are no harder to evaluate than inverse integral cotangents; those who prefer to deal only with the latter may mentally substitute the expression {2[N] - [N(N²+3)/2]} for "[N/2]" (in this example: {2[1011] - [516683682]}). However, we have no plans to extend the database to include reductions of inverse fractional cotangents involving denominators greater than 2.

The database comprises a number of text files - the first of these contains the reductions of all inverse cotangents [N] and [N/2] for which the largest prime factor of (N² + 1), or (for half-integers) of (N² + 4), is at most 97; each of the remaining files corresponds to a single irreducible whose associated prime exceeds 97. Techniques for populating files in this database are described in the software page - in general (as is to be expected), the higher the associated prime, the larger the file.

The database file corresponding
to the irreducible [49] may be taken as an example. Since (49²
+ 1) = 2.1201, 1201 is the associated prime. This file contains
(only) reductions of inverse cotangents [X], where X = N (integer)
or X = N/2 (half-integer), and 1201 is the ** largest**
prime factor of (N² + 1) or (N² + 4) respectively; every
one of these reductions (of which there are currently over 18,000)
is certain to include a multiple of [49]. Reductions held in database
files corresponding to irreducibles whose associated prime is

The software includes programs which can search this database for all reductions involving only a wholly- or partly-specified set of irreducibles, and (treating these reductions as simultaneous linear equations, as in the example above) can then attempt to eliminate all the irreducibles except [1] from the equations in increasing sequence (i.e. [2], [4], [5], [6],...). This process, if successful, leads to an arccotangent identity with [1], or a positive multiple of [1], on the LHS. The irreducibles eliminated are referred to as the "eliminated set" (sometimes some irreducibles cannot be eliminated, but feature in the resultant identity).

Many of the identities in our lists are present either for their historical interest, or as a record of the progress made by the editors in their search for new identities. Every identity has a "measure" which is the sum of the reciprocals of the logarithms, to base 10, of the cotangent values (integers and half-integers) it includes - the smaller the measure, the better. We also use an identity's measure, to 5 decimal places, to act as a distinguishing identifier for it - this never seems to lead to confusion.

Our current "best" identity has measure 1.26579.

Any serious attempt to compute PI using arccotangent identities will usually rely on a pair of mutually-checking identities, the results only being acceptable if no significant discrepancies are detected. To minimise computation time, the pair should share as many cotangent values as possible, having coefficients which (after "normalisation" relative to the coefficient of [1]) must be different in the two identities. Examples are listed on the "Self-checking pairs" page; the "Compound measure" shown on the line preceding each pair, used to index the pair's efficacy, is the sum of the reciprocals of the logarithms of all distinct cotangent values (DCVs), in effect the sum of the measures of the two identities, but with the contributions of shared cotangent values counted only once; pairs are grouped according to the number of DCVs they contain, making it easier for users with a particular number of PCs at their disposal to choose a suitable pair to use in a computation.

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To determine whether a particular
collection of irreducibles can provide us with a worthwhile self-checking
pair, we search the database for all reductions involving only
these irreducibles, and then attempt to find within those the
"best" linearly-independent set of reductions from which
all of these irreducibles, **as well as [1],** can be eliminated
- i.e. we look for a linear sum of "reducible" inverse
integer and half-integer cotangents equating to 0. If any single
"reducible" is then omitted from this set of values,
an identity should be obtainable from the values remaining, any
two such identities constituting a "self-checking" pair.
All such "pairs" will have the same "best"
Compound measure - by convention, we choose the two identities
obtained by omitting in turn the two lowest cotangent values in
the set, so that the resulting pair of identities have the least
measures obtainable.

Our current "best" mutually-checking pair of identities has Compound measure 1.81462.

To access the classified lists of identities, follow a link to "Identity lists". To access the list of self-checking pairs of identities, follow a link to "Self-checking pairs". To explore some of the byways, follow links in the top line of this page.

Note: References to "N-digit" identities below imply that, in any such identity, the smallest cotangent value (or its numerator, in the case of a half-integer cotangent) has N decimal digits.

- New 3-DCV self-checking pair with Compound measure 4.62989.

- In 6-digit identity with measure 1.61816, associated Compound measure corrected to 1.85000.

- In 2-digit identity with measure 1.74096, coefficients of both last 2 terms changed from '-4' to '+4'.
- In 2-digit identity with measure 1.78220, coefficient of [110443] changed from '+6' to '+12'.
- In 8-digit identity with measure 1.71753, missing ']' appended to 14th term.
- In 10-digit identity with measure 1.82151, representation of [36523075989/2] corrected.
- In 12-digit identity with measure 1.92755, '2]' appended to 3rd term.
- In 12-digit identity with measure 1.93336, '4' appended to coefficient of 4th term (becomes '+4765038574').
- In 13-digit identity with measure 2.01249, representation of [43777069138111/2] corrected.
- In 13-digit identity with measure 2.01560, representation of [17857722814749/2] corrected (spurious '{2' removed).
- In 'Self-checking pairs', identity with measure 1.65426, coefficient of 6th term should be '-67595' (minus sign was missing).
- In 'Self-checking pairs', identity with measure 1.69382, change sign of coefficient of last term (becomes '+13261').
- In 'Self-checking pairs', identity with measure 1.70951, change sign of coefficient of last term (becomes '+26522').

- In 2-digit identity with measure 1.65181, sign preceding term [18677233307] changed from '-' to '+' .
- In 9-digit identity with measure 1.73500, coefficient of [1432278547] changed from '+54876' to '+5487606'.

- 4th cotangent value in "best" 2-digit identity corrected to [9916207] (measure unchanged).

- New 6-digit identity (1.65027) and two new 5-digit identities (1.53702 (new "best"), 1.58108).

- Dates associated
with Euler's two 3-term identities revised following research
by Amrik Singh Nimbran in the "Euler Archive" <
__http://www.math.dartmouth.edu/~euler__>.

- New 9-digit identity (1.74866), member of new "best" 18 DCV self-checking pair (Compound measure 1.88600).
- New 8-digit identity with measure 1.73223.

- Further minor revisions to text on this page (above) relating to database of reductions.

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- Revisions to text on this page (above) relating to database of reductions.
- Minor changes
to Software page - including enhancement of process to generate
new "outsize" half-integer reducibles with associated
primes up to 1201; these can also (more frequently than one might
expect) be obtained by "differencing" integer reducibles
whose associated primes
*exceed*1201.

- New 12-digit identity with measure 1.92909.

- New 14-digit identities with measures 2.08477, 2.08500, 2.09165, 2.09335, 2.09414, 2.09485, and 2.09539.
- New 31-DCV self-checking pairs with Compound measures 2.16051 and 2.16117.

- The "Software" page has been revised, and now includes an outline description of a recently-completed program which enables new "outsize" reducible half-integer cotangent values to be generated. This program has been applied with some success; when all the results have been gathered in, improvements to several reductions displayed on the "Identity List" pages for (probably) 12-, 13- and 14-digit identities (and in some cases the "Self-checking pairs" page) will be incorporated.
- A reference has been added on the LOGOUT page to Amrik Singh Nimbran's article, entitled "On the derivation of Machin-like Arctangent Identities for computing Pi", recently published in the Indian journal "The Mathematics Student".

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- Five new 6-digit identies discovered by Amrik Singh Nimbran (29May11) with measures 1.61816, 1.65318, 1.65587, 1.65629, and 1.66027 (a sixth, with measure 1.66333, also incorporated in a new 13-DCV self-checking pair, below).
- Three concurrently-discovered new 13-DCV self-checking pairs, incorporating some of the above, with Compound measures 1.85000, 1.85894 and 1.86616.
- Four new 12-digit identities discovered with measures 1.90181 (ASN, 16Jun11), 1.93217 (ASN,21Jun), 1.90536 and 1.91506.
- Two new 5-digit identities (25Jun11) with measures 1.62115 and 1.63062.
- Two new 14-digit identities with measures 2.09537 (17Jun11) and 2.09750 (25Jun11).
- New 31-DCV self-checking pair with Compound measure 2.17215.

- "Software" page updated to record the fact that all files in our database of reductions for irreducibles up to [49] (i.e. associated with primes up to 1201) have been completed for LHS integer cotangent values up to [9999 99999 99999] and half-integer values up to [19999 99999 99999/2].

- 3-digit identity list tidied considerably. It will be observed that, in certain cases, multiple identities derived by "tweaking" from a single original have been listed, for the sake of interest ; now, for each "original", only the measure of the best such "derivative" is included in the "top 20 measures" list for the category.

- Website title,
and wording of above introduction, revised to remove emphasis
on inverse
cotangent terms in identities.*integral*

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- Revised wording on "Identity Lists" page to clarify categorisation of identities with more than four terms.

- New "natural" half-integer convention applied to all identity lists.

- The "Tweaking" page has been more drastically edited, to reflect changes to the associated software, and also to announce some resulting successes.
- Many of the resulting changes involve a half-integer term [3375905320682366575989 / 2]. Since the cube of the numerator of this fraction takes up an inordinate amount of space, it has been decided to abandon the rather artificial practice of representing half-integer terms as pairs of integers (this change is on-going).

- Layout of identities (on "Identity lists" page) updated to introduce notion of "back-formed" identities, identified by discovery date shown as "(***)". 2-digit, 3-digit and 4-digit identity pages updated to conform.

- A note has been inserted on the "Identity Lists" page to clarify the significance of asterisked references to other identities.

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- A selection of Amrik Singh Nimbran's identities, discovered during 2007-8 and recently sent to us, appear in the 3- and 4-digit lists. We are grateful for these - more of them are likely to be included after checking; they have stimulated us to generate new identities ourselves.

- "Computing PI" added to title.

- Page added describing programs used to perform evaluation of PI featured on this website.

- Preparation for revival of website (lapsed since end July 2008) with URL changed to <http://www.machination.eclipse.co.uk>.

- Our attention has recently been drawn to Ian Tweddle's 1991 paper "John Machin and Robert Simson on Inverse-tangent series for PI" (v. detailed reference on LOGOUT page). This paper, and its references, show convincingly that Machin submitted a paper to the Royal Society in 1705 or 1706 which described not only the four 2-term identities listed herein (including his "famous" one), but also a 3-term identity hitherto attributed to Loney (1893); moreover Simson (1687-1768) can now be credited with describing, in 1723, the 3-term identity documented by Klingenstierna in 1730. These attributions have accordingly all been revised.