Rapid computation of Bernoulli and related numbers

Bernoulli numbers and similar arithmetic objects have long been of interest in mathematics. Historically, people have been interested in different recursion formulae that can be derived for the Bernoulli numbers, and the use of these recursion formulae for the calculation of Bernoulli numbers. Some of these methods, which in the past have only been of theoretical interest, are now practical with the availability of high-powered computation. This poster explores some of these techniques of deriving new recursion formulae, and expands upon these methods. The main technique that is explored is that of "multisectioning'. Typically, the calculation of a Bernoulli number requires the calculation of all previous Bernoulli numbers. The method of multisectioning is such that only a fraction of the previous Bernoulli :numbers are needed. In exchange, a more complicated recursion formula, called a "laeunary recursion formula", must be derived and used. 1 H i s t o r y Bernoulli numbers were first introduced by Jacques Bernoulli (1654-1705), in the second part of his treatise published in 1713, Are conjectandi ("Art of Conjecturing"). At the time, Bernoulli numbers were used for writing the infinite series expansions of hyperbolic and trigonometric functions [2]. Van den Berg was the first to discuss finding recurrence formulae for the Bernoulli numbers with arbitrary sized gaps (1881) [4]. Ramanujan showed how gaps of size 7 could be found, and explicitly wrote out the recursion for gaps of size 6 [1, 4, 5]. Lehmer in 1934 extended these methods to Euler numbers, Genocchi numbers, and Lucas numbers (1934) [4], and calculated the 196-th Bernoulli number. These methods to find new recurrences for Bernoulli numbers, and numbers related to the Bernoulli numbers were automated in Maple, and discussed in a much more general setting for my thesis [3].