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Pippenger Product

The Pippenger product is an unexpected Wallis-like formula for given by(1)(OEIS A084148 and A084149; Pippenger 1980). Here, the th term for is given by(2)(3)where is a double factorial and is the gamma function.

Pickover's Sequence

Pickover's sequence gives the starting positions in the decimal expansion of (ignoring the leading 3) in which the first digits of occur (counting the leading 2). So, since , the first digit "2" of occurs at position 6. Continuing, the sequence is given by 6, 28, 241, 11706, 28024, 33789, 1526800, 73154827, ... (OEIS A090898).Conversely, consider the sequence formed by the expansion of (ignoring the leading 2) in which the first digits of occur (counting the leading 3). So, since , the first digit "3" of occurs at position 17. Continuing, the sequence is given by 17, 189, 856, 17947, 53238, 1436935, 5000482, ... (OEIS A115234).

E Pi

e Approximations

An amazing pandigital approximation to that is correct to 18457734525360901453873570 decimal digits is given by(1)found by R. Sabey in 2004 (Friedman 2004).Castellanos (1988ab) gives several curious approximations to ,(2)(3)(4)(5)(6)(7)which are good to 6, 7, 9, 10, 12, and 15 digits respectively.E. Pegg Jr. (pers. comm., Mar. 2, 2002), found(8)which is good to 7 digits.J. Lafont (pers. comm., MAy 16, 2008) found(9)where is a harmonic number, which is good to 7 digits.N. Davidson (pers. comm., Sept. 7, 2004) found(10)which is good to 6 digits.D. Barron noticed the curious approximation(11)where is Catalan's constant and is the Euler-Mascheroni constant, which however, is only good to 3 digits.

e Digits

The constant e with decimal expansion(OEIS A001113) can be computed to digits of precision in 10 CPU-minutes on modern hardware. was computed to digits by P. Demichel, and the first have been verified by X. Gourdon on Nov. 21, 1999 (Plouffe). was computed to decimal digits by S. Kondo on Jul. 5, 2010 (Yee).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 2, 252, 1361, 11806, 210482, 9030286, 3548262, 141850388, 1290227011, ... (OEIS A224828).The starting positions of the first occurrence of in the decimal expansion of (including the initial 2 and counting it as the first digit) are 14, 3, 1, 18, 11, 12, 21, 2, ... (OEIS A088576).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 6, 12, 548, 1769, 92994, ... (OEIS A036900), which end at digits 21, 372, 8092, 102128, ... (OEIS A036904).The digit sequence 0123456789..

e Continued Fraction

The simple continued fraction representations of given by [2; 1, 2, 1, 1, 4, 1, 1, 6, ...] (OEIS A003417). This continued fraction is sometimes known as Euler's continued fraction. A plot of the first 256 terms of the continued fraction represented as a sequence of binary bits is shown above.The convergents can be given in closed form as ratios of confluent hypergeometric functions of the first kind (Komatsu 2007ab), with the first few being 2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, ... (OEIS A007676 and A007677). These are good to 0, 0, 1, 1, 2, 3, 3, 4, 5, 5, ... (OEIS A114539) decimal digits, respectively.Other continued fraction representations are(1)(2)(3)(Olds 1963, pp. 135-136). Amazingly, not only the continued fractions , but those of rational powers of show regularity, for example(4)(5)(6)(7)A beautiful non-simple continued fraction for is given by(8)(Wall 1948, p. 348).Let the continued fraction of be denoted..


The constant is base of the natural logarithm. is sometimes known as Napier's constant, although its symbol () honors Euler. is the unique number with the property that the area of the region bounded by the hyperbola , the x-axis, and the vertical lines and is 1. In other words,(1)With the possible exception of , is the most important constant in mathematics since it appears in myriad mathematical contexts involving limits and derivatives. The numerical value of is(2)(OEIS A001113). can be defined by the limit(3)(illustrated above), or by the infinite series(4)as first published by Newton (1669; reprinted in Whiteside 1968, p. 225). is given by the unusual limit(5)(Brothers and Knox 1998).Euler (1737; Sandifer 2006) proved that is irrational by proving that has an infinite simple continued fraction (; Nagell 1951), and Liouville proved in 1844 that does not satisfy any quadratic equation with integral coefficients (i.e., if it is..

Gelfond's Constant

The constant that Gelfond's theorem established to be transcendental seems to lack a generally accepted name. As a result, in this work, it will be dubbed Gelfond's constant. Both the Gelfond-Schneider constant and Gelfond's constant were singled out in the 7th of Hilbert's problems as examples of numbers whose transcendence was an open problem (Wells 1986, p. 45).Gelfond's constant has the numerical value(1)(OEIS A039661) and simplecontinued fraction(2)(OEIS A058287).Its digits can be computed efficiently using the iteration(3)with , and then plugging in to(4)(Borwein and Bailey 2003, p. 137).

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