Lehmer's Constant
N.B. A detailed on-line essay by S. Finchwas the starting point for this entry.
Lehmer (1938) showed that every Positive Irrational Number 
has a unique infinite continued cotangentrepresentation of the form
where the
s are Nonnegative and 
The case for which the convergence is slowest occurs when the inequality is replaced by equality, giving
and
for
. The first few values are 
are 0, 1, 3, 13, 183, 33673, ... (Sloane's A024556), resulting in the constant |  |  | |
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(Sloane's A030125).
is not an Algebraic Number of degree less than 
, but Lehmer's approach cannot show whetheror not is Transcendental.See also Algebraic Number, Transcendental Number
References
Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/lehmer/lehmer.html Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 29, 1983. Lehmer, D. H. ``A Cotangent Analogue of Continued Fractions.'' Duke Math. J. 4, 323-340, 1938. Plouffe, S. ``The Lehmer Constant.''http://www.lacim.uqam.ca/piDATA/lehmer.txt. Sloane, N. J. A.A024556 andA030125 in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.
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