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ELIMINATED SETS

Coded notation for leading irreducibles in eliminated set:
{A} = [2][5]
{B} = [2][4][5]
{C} = [2][4][5][12]
{D} = [2][4][5][6][12]
{E} = [2][4][5][6][9][12]
{F} = [2][4][5][6][9][12][23]
{G} = [2][4][5][6][9][11][12][23] ("Gauss's set")
{H} = [2][4][5][6][9][11][12][23][27]
{I} = [2][4][5][6][9][11][12][23][27][34]
{J} = [2][4][5][6][9][11][12][22][23][27][34]
{K} = [2][4][5][6][9][10][11][12][22][23][27][34]
{W} = [2][4][5][6][10][11][12][22][23][34] ("Arndt's set")
    = {K -9 -27} (note convention used here to indicate exclusion of specified irreducibles)

A table showing the prime number associated with each irreducible is included in the pages on Todd's reduction process (follow link from Home Page).

NOTES:

• The absence of an eliminated set usually means either that the identity is a reduction (of the term with the largest cotangent value) in its own right - many of Størmer's 3-term identities are of this type; or that it is the result of "tweaking" an original identity by the substitution of a "triplet" for a pair of terms.
  
• An entry such as "[6&69]" in an eliminated set implies that, in this case, the elimination of the irreducible [6] from the linear equations caused the simultaneous elimination of [69].
  
• When an eliminated set accompanies one of the identities listed, it does not necessarily follow that the Machination software would generate that identity if presented with that particular set of irreducibles; it simply implies that those irreducibles, and no others, occur in the reductions of the non-irreducible terms in the identity (however when the pair of integer terms [N] and [N(N²+3)/2] is "substituted" for a half-integer [N/2], irreducibles associated only with [N] and [N(N²+3)/2] - i.e. not associated with other terms in the identity - will not normally be included in the eliminated set).
  
• Irreducibles in parentheses in the eliminated set, e.g. ([107]), denote those which, although present in one or more reductions, could not be eliminated, and are perforce included in the identity.

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