*N.B. A detailed on-line essay by S. Finch
was the starting point for this entry.*

Lehmer (1938) showed that every Positive Irrational Number has a unique infinite continued cotangent
representation 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

(Sloane's A030125). is not an Algebraic Number of degree less than , but Lehmer's approach cannot show whether or not is Transcendental.

**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 and A030125 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

© 1996-9

1999-05-26