## Lehmer's Constant

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.