Several years ago the second author, playing with di?erent “recognizers of real constants,” e.g., the LLL algorithm, the Plou?e inverter, etc., found the following formula empirically. Let pn/qn denote the nth convergent of the continued fraction of the constant e. Then X n0 |qne pn| = e 4 0 @1 + 10X n0 (1)n (n + 1)!(2n2 + 7n + 3) 1 A . The purpose of the present paper is to prove this formula and to give similar formulas for some powers of e.
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